Is it really better for the environment if I take the stairs as opposed to a lift?
I follow your analysis but not your conclusion. You are missing two key points, which as far as I can see might outweigh all the other points you mention:
You must compare the efficiency of the electric engine running the lift with your body-"engine".
You must compare the "fuel" that the electric engine and your body-"engine" use.
To the first case, all your other points pale and are negligible, if an old inefficient engine is in use, or opposite if you as the person walking up the stairs have a hard an inefficient time doing so.
To the second case, what if the electric engine is run by the roof solar panel, geothermal heating or a local wind turbine? Then the elevator's impact becomes negligible*. On the other hand, if you eat, drink and gain nutrition in an environmentally sustainable manner**, then your impact might be considered negligible.
These thermodynamic and fuel-production factors*** are, as far as I can see, more important than your presented analysis, which is based purely on mechanical energy.
* Unless we include an analysis of the lifetime, production and disposal of the solar cell, geothermal plant for wind turbine, in which case they will never win with current development technologies. We with then also have to consider other uses of these sources, and the math quickly becomes large and complicated.
** Home-grown vegetables etc.
*** I'm sure we can find more such factors if we dig deep into the details of the lift technology and human physiology. The environmental-impact-math will also highly depend on how we estimate and evaluate indirect factors that may make all of your and my points moot, some of which are mentioned in comments and other answers, such as the energy storing capability of a lift's counterweight or such as the question of if the lift will be implemented and in use regardless of what you choose to do due to regulation and accessibility for the disabled/elderly.
"Better for the environment" is not an objective scientific quantity. To do the comparison you need something more specific like "the one that releases the less CO2".
The lift, depending on its type, releases (indirectly) a quite stable amount of CO2 each time someone uses it. It depends on too many factors to know what it is for this specific lift, but considering such a sign was displayed it is probably not a super-clean model.
When you use the stairs your body indeed uses more energy, but this doesn't necessarily means you will eat more: the human body is wasteful and it requires a significant amount of daily effort to reach the level at which you start to eat more. This means that in most cases climbing those stairs will have literally zero effect on environment.
There are two conflicting ways to think about this, both with there own adherents:
The work you do is based on renewable energy: food. Even the best elevator uses some electricity, which has environmental impact. Take the stairs!
Any building with elevators has air handling, and perhaps air cooling. The heat you add going up stairs takes more energy to extract than what’s needed to run the elevator. Take the lift.
Both of those are over-simplified, which makes the arguments go on and on.