Modern Papin engine depiction
Left: Original 1690 Papin engine diagram. [1] Right: modern drawing of 1712 Papin engine diagram. [2]

Left: an 1690 sketch of the Papin engine, according to which water is placed in the cylinder, the piston is then heated from the outside, thus turning the water into steam, which forces the piston up, which is then latched into position, by notch E; the piston is then cooled, which makes a powerful vacuum inside, such that when the latch is turned, the downward force of the piston can work the rope and pulley, so to lift 60 pounds, using a 2.5-inch diameter cylinder. [3] Right: an annotated version using his 1690 journal publication illustration of his new steam engine.

“Since it is a property of water that a small quantity of it, turned into vapour by heat, has an elastic force like that of air, but upon cold supervening is again resolved into water, so that no trace of the said elastic force remains, I readily concluded that machines could be constructed wherein water, by the help of no very intense heat, and at little cost, could produce that perfect vacuum which could by no means be obtained by the aid of gunpowder.”

— Denis Papin (1690), "A New Method to Obtain Very Great Motive Powers at Small Cost”

“In the Acts for 1690, p. 410, Papin published an addition to his former paper. He says, that finding it impossible to make a complete vacuum in the cylinder by means of gunpowder, he had endeavored to obtain his object by means of water, which, he says, 'has the property when changed into vapor to spring like the air' and afterwards to recondense itself so well by cold, that there remains “no appearance of this force or spring.” The machine which he proposed is a cylinder, AA, made of thin metal, and fitted with a piston B, which can slide freely up and down in the cylinder. A small quantity of water is put into the bottom of this cylinder, and the piston B put down so as to touch the water, the air being expelled through a hole in the piston, which is afterwards plugged up by a plug M. A fire is applied beneath the bottom of the cylinder to heat the water within, and make it boil, which will soon be done, the bottom being made of very thin metal. The water is then changed into vapor, which exerts so strong a pressure beneath the piston, that it surmounts the pressure of the atmosphere, and pushes the piston upwards to the top of the cylinder, and then a latch E falls into a notch in the stem H of the piston, to prevent it from going down again. The fire must now be taken away from beneath the cylinder, and the vapors in this thin cylinder soon recondense themselves into water by cold, and leave the cylinder entirely empty of air. In this state, the machine is ready to exert its force, for by only releasing the latch E, the piston will be pressed down into the cylinder by the weight of the atmosphere which presses upon its upper surface, whilst there is O no air beneath it to resist its descent, and a rope L being fastened to the stem of the piston H, may be conducted over pulleys TT, and applied to raise weights. The principle of the atmospheric steam-engine is suggested in this rude machine, which was afterwards perfected ; the author states that he tried the experiment with a cylinder 2.5 inches diameter, which he found capable of raising 60 pounds (that is = 12.25 pounds per square inch) and that it could be made to repeat its action once a minute; from this data he calculated, that a cylinder of little more than two feet diameter, and four feet in height, would have sufficient force to raise a weight of 8000 pounds every minute to a height of four feet; this is 32,000 pounds raised one foot per minute, which is nearly a modern horse-power, or equal to the force of 8.5 men. Papin states his invention to be applicable to draw water from mines, to throw bombs, and to row vessels against wind and tide.”

— John Farey (1827), A Treatise on the Steam Engine (pgs. 97-98) [3]

The following shows how "Papin engine" (1688/1690) inspired or rather became the design template behind the: Savery engine [Miner’s friend] (1698), Newcomen engine (1712), and Watt engine (1765-96): [5]

Now, the actual historical details of how Papin's 1688 gunpowder engine and 1690 steam engine designs influenced the minds of Thomas Savery (1698) and his engine (Savery engine) and Thomas Newcomen and his engine (Newcomen engine) becomes murky, if not an semi-unsolved problem in the history of the steam engine.

Firstly, that all of Papin's engine designs were openly published for others to read about and work on, such as the German periodical Acts of Erudite (1688 and 1690); in the form of a book in French entitled Collection of Various Pieces Touching Some New Machines; and most importantly that summarizes of his engine models were read to the Royal Society in 1686 and 1687, which both Savery and Newcomen were connected with. Moreover, Papin, in 1676 to 1679, was an assistant to Robert Hooke, the curator of experiments of the Royal Society, and that in 1682, some of the Royal Society's dinners were being cooked in Papin's digestor. We also know that Hooke was in communication with Thomas Newcomen and had told him about Papin’s engine designs. We also know that Savery and Newcomen resided 15-miles from each other.

Now, as to Savery, details are wanting, but we do know that in 1698 he obtained a patent for his new engine, albeit with description or technical details, and on 14 Jun 1699, Savery demonstrated a scale model of his new steam engine to the Royal Society.

The following is one conjecture on the Papin-Savery connection, namely that Savery learned of Papin in the years when 1686 to 1690, during which time Papin’s theoretical engine designs came under the attack of Robert Hooke, who in 1675 had had published a cypher-coded secret description his own vacuum-based heat engine theory, which caused Papin to retreat for four years, and that Savery observed and learned from this attack:

“Thomas Savery filed the first patent in 1698 for a steam engine that allowed water to be raised by the driving force of fire, the first copy of which was built the same year and presented in 1699 to the Royal Society of London and the King. This patent was quite particular in that it provided that any fire machine that was manufactured or installed in England would be confiscated and placed under the control of Thomas Savery. Moreover, this patent imposed secrecy on Savery's machine, and it was not until 1704 that a spy copied this description in Hanover and transmitted it to Leibniz. Thomas Savery was a former mine worker. As a marine captain and engineer, he became aware of the work of Denis Papin, who was Robert Hooke's assistant in 1679. The latter had presented a series of criticisms of Denis Papin's work at a meeting of the Royal Society of London, widely publicized criticisms that pushed Papin to abandon his research until 1694. For Thomas Savery, these comments, the reversal of the members of the Learned Societies, his desire to quickly file a patent and the pressures of the English mining managers, not to mention the time needed to create a new machine that would not have the defects reported in Papin's, prompted him to choose an intermediate strategy, a strategy that consisted of developing a revised and less complex steam engine (without cylinder or piston) but specialized for pumping water from the mines.”

— Michel Vigezzi (2019), World Industrialization: Shared Inventions, Competitive Innovations, and Social Dynamics (pg. #)

In sum, although exact historical verification is wanting, baring Vigezzi's conjecture (above), it is HIGHLY-unlikely that Savery was ignorant of Papin in 1698 when he applied for his patient, given the nearly to exact overlap of his design and Papin's design, both having a hot body, cold body, and vacuum chamber.

1707 | Engine
In 1705, Gottfried Leibnitz sent Papin a description of the Savery engine, which promoted Papin to suggest improvements to Savery’s design, which resulted in the following 1707 steam engine model: [6]

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“In 1688, Papin published in the Acts of Erudite, a method of prolonging the motion of water-wheels to a great distance, by drawing air through pipes; also a project for a new use of gunpowder. Papers on both these subjects had been read to the Royal Society in 1686 and 1687. In the Acts of Erudite for 1690, is a further addition to his former paper on the use of gunpowder. In this he proposes to use steam, and here is the origin of the steam-engine. These three papers, which are in Latin, contain the sum of Papin's inventions, and are certainly of great merit. He afterwards republished them in French, with some additions, in a small book, entitled: Collection of Various Pieces Touching Some New Machines (Recueil de diverse Pièces touchant quelques nouvelles Machines), par D. Papin, Cassel, 1695.” It is an interesting publication, and does the author great credit.”

— John Farey (1827), A Treatise on the Steam Engine [3]

“Papin's project of 1687, for using gunpowder, is of no real utility: indeed, he was sensible that he had not succeeded, and said he published it only to excite the learned of other countries to perfect it. (See the Acta Eruditorem for 1688, p. 497.) He proposed to employ a cylinder with a piston fitted into it, and to the rod of this piston the weights which were to be raised were applied by a rope conducted over pulleys, and they were to be lifted by the pressure of the atmospheric air upon the piston, when the air was exhausted from the cylinder. Thus far, he followed Otto Guericke; but instead of exhausting the cylinder by an air pump, Papin proposed to drive out the air by firing off a small quantity of gunpowder in the inside of the cylinder. The piston had a large aperture in it, covered by a valve opening upwards; and he thought that the sudden blast of flame produced by the explosion of the gunpowder would lift up the valve, and drive out the air at the aperture; but when the flame ceased, the valve would fall, and prevent the return of the air into the cylinder, so as to leave it exhausted, and then the pressure of the atmosphere on the upper surface of the piston not being balanced by any air beneath, would press the piston down with great force. Papin speaks of two models of this machine which he tried; one of them five inches diameter and 16, inches high; but he says he could never succeed in expelling the whole of the air from the inside of the cylinder: about one-fifth always remained, and that fifth, as he says, reduced the force to one-half of what it would have been if he could have expelled the whole of the air.”

— John Farey (1827), A Treatise on the Steam Engine [3]

“I find it almost impossible to believe that Newcomen never knew of Papin’s steam engine cylinder, at least by hearsay. Papin published sever other papers (3,4,5) and maybe there was someone at Dartmouth, one of the gentry perhaps, who could read French, if not Latin, and gave Newcomen access to Papin’s work.”

— Joseph Needham (1987), Science and Civilization in China (pg. 559)

“The similarities between Newcomen's steam engine design and Papin's first piston-and-cylinder engine also suggest that Newcomen might have seen Papin's articles published by the Royal Society.”

— Anon (2012), Engineers [6]