Does 245 count floors?

nope, sorry

The 245 doesn't have a barometric altimeter

I bought my wife a 245M yesterday and when I mentioned the lack of floor count she said that it was rubbish anyway. My personal view after a Fenix 3, Fenix 5 and now a 945 is that it isn't particularly accurate and a novelty at best, especially if you are spend a much of your day inside and outside of buildings. I have often got my stair goal when i've not been near any and missed it when i've been up and down them all day inside a building.

I completely agree.  I temporarily owned a 935 and it bothered me that after runs I would see an increase on the number of floors climbed.  I was running you know, not going up floors but somehow the 935 was not smart enough to tell the difference...

But don't hold your breath in general with Garmin.  The more you "poke" at "features" like that, the more you realize they are gimmicky things to get you to buy the unit but the actual implementation and real-life use is many times lacking.

Why does it lack a barometric altimeter?  Even my cheap Fitbit has it.  Why would Garmin leave it out??

Because Garmin chose not to put one in this watch.  Does your "cheap Fitbit" do everything your 245 does?  If it does, I would say to you to switch back.

The vivoactive HR doesn’t have an altimeter/barometer but counts stairs fairly accurately with an accelerometer. 245 should be able to do the same but doesnt

The vivoactive HR does have a barometric altimeter.

I have a fenix 5x.   Novelty is a perfect description.   However I probably get more exercise climbing up and down Stairs in my house just trying to get my watch to notice.   Swinging arms does seem to help.