I track my flights of stairs and have a goal of averaging 25 flights per day since I got the s3 last fall. I have posted here on this question at times and have tried to figure it out, but have questions more than answers too.
It is definitely not just the barometric reading. There has to be use of the GPS and other sensors in the logic, and also smoothing. For your stairs inside, the likely explanation is that it is missing the top and the bottom of the stairs and as a result of smoothing and other error correction, you are not getting credit. So if you go up the stairs and then walk around the 2nd floor, and then go down the stairs and walk around the 1st floor, you are much more likely to get credit, because the other sensors will count your movement and position at the higher and lower pressure. In some way at least.
I have several apps to measure altitude and pressure. One simple one is called Up High. It just gives you a real time reading of the pressure and altitude. it bounces around by 1-3 feet just sitting still at a desk, and sometimes up to 5 feet. So there is a margin of error in the barometer that is a significant part of 1 flight of stairs. Another example of this is that there needs to be minimum number of stairs to ever count: I stay in a house that has 3 stairs in and out; I can go in and out of the house 30x in a day, but will end up with basically 0 flights for entire the day, so those short flights of stairs never amount to anything.
I really like and recommend the app WorkOutdoors. The developer posts here and has explained his app a lot. WorkOutdoors is the only I know that gives you the option for a real time reading of flights of stairs as you climb them. Outside, on a hill, I can watch it count both elevation and flights of stairs as I do them. But workoutdoors does use GPS (at least in part), and it will at times round off the top or bottom of the hills as part of its logic smoothing out changes in elevation. So that app shows that happening in real time as it happens. Workoutdoors does not appear to track stairs indoors very well or at all, but I should test that more (using GPS logic may prevent that).