Modern Times have Missed the point mightily

The spelled out property of reversibility is "strict reversibility" specifically when work is done on the machine, and equal amount of heat is produced. This speaks of the mechanical effect by and on the expanding medium. This exact equivalence is a property of vapor, and so of all vapor engines.

Carnot and Kelvin both state many times and many ways this is a property of both steam engines and air engines. Not just Carnot Cycle Cylinders. How this over the past hundreds of years came to be associated with the "Carnot cycle" as being unique or different from other engine cycles is a mystery. The giants of Thermodynamics and Heat engines spelled it out, and we missed the point. At one point, Lord Kelvin states the properties and calculations are "applicable to any conceivable kind of thermodynamic engine."

The other property must also be met. A "perfect" engine must not lose its heat based on diffusion (including exhaust heat, which is eventually diffused). This is the property of Enclosability. An Enclosable heat engine, will not increase the heat of its pressure envelope, in a steady state even if operated in a finite pressure envelope.

In fairness, Carnot and Kelvin's work was written presuming there is heat left to drain out after work has been done. So the premise of a perfect engine would not be that it drained no heat, but that it drained the heat at the temperature of the "cold source". So that the heat had already "fallen" all it could.

This is equivalent to statement: For either heat model. In a perfect engine, no temperature drop occurs except for in creation of work. This is equivalent to not dispersing any heat at a temperature above the Pressure Envelopes ambient temperature, or in the waterfall model, the "drained" heat must be at the same temperature as the cold source. It is the latter that Lord kelvin describes in his version of the Carnot Cycle using both steam and air. However, both Kelvin and Carnot fail to recognize the limit the pressure envelope creates, that the lower temperature may be unreachable, unless negative work is done to force expansion of the volume past the point that pressures are equalized. Their model was an equal amount of heat left the hot source, and "fell" to the cold source. Had they realized if the air or steam reaches the ambient temperature of the cold source, that NO heat is dispersed, history might have been different.

So the premise in "Reflections" of the concept of "materiality of Heat" implies the necessity of a "heat drain" or cold source, begins the chain of mistakes in analyzing the behavior and limits of heat energy conversion.

"Reflections" page 11 editors note, on materiality of heat, 
chapter THE WORK OF SADI CARNOT
  Carnot was the first to declare that the maximum
  of work done by heat, in any given case of application
  of the heat-energy, is determined solely by
  the range of temperature through which it fell in
  the operation, and is entirely independent of the
  nature of the working substance chosen as the
  medium of transfer of energy and the vehicle of
  the heat. His assumption of the materiality of
  heat led, logically, to the conclusion that the
  same quantity of heat was finally stored in the
  refrigerator as had, initially, left the furnace, and
  that the effect produced was a consequence of a fall
  of temperature analogous to a fall of water; but,
  aside from this error which he himself was evi-
  dently inclined to regard as such, his process and
  argument are perfectly correct.

Both Kelvin and Carnot's description of the Carnot Cycle acknowledge that heat is leaving the system during the "cold" compression cycle. Consequently, the Carnot cycle fails the enclosability test, which is the first and most difficult criteria of not dissipating any of the heat which fuels (not moves) the engine.

"Reflections" pg 56-57, Sadi Carnot on maximum efficiency of operation
chapter MOTIVE POWER OF HEAT.

  Since every re-establishment of equilibrium in
  the caloric may be the cause of the production of
  motive power, every re-establishment of equilibrium
  which shall be accomplished without production of
  this power should be considered as an actual loss.
  Now, very little reflection would show that all
  change of temperature which is not due to a change
  of volume of the bodies can be only a useless reestablishment
  of equilibrium in the caloric.* The
  necessary condition of the maximum is, then, that
  in the bodies employed to realize the motive power
  of heat there should not occur any change of temperature
  which may not be due to a change of
  volume.

page 161 Lord Kelvin, The same calculations apply to real air engines

(paragraph) 28...
  Regnault's very accurate experiments show that
  the deviations are insensible, or very nearly so, for
  the ordinary gases at ordinary pressures ; although
  they may be considerable for a medium, such as
  sulphurous acid, or carbonic acid under high pressure,
  which approaches the physical condition of a
  vapor at saturation ; and therefore, in general, and
  especially in practical applications to real air-engines,<<<
  it will be unnecessary to make any modification in
  the expressions.

(paragraph) 29.* Either the steam-engine or the air-engine,
  according to the arrangements described above,
  gives all the mechanical effect that can possibly be
  obtained from the thermal agency employed. 
  
  (footnote) *
  This paragraph is the demonstration, referred to above,
  of the proposition stated in 13, as it is readily seen that
  it is applicable to any conceivable kind of thermodynamic
  engine.