Hello, Alexey. I have read only the previous page of this long thread of yours and don't know all the details of the discussion, so I hope you won't judge me too hard if what I'm going to say was already said.
The examples in your quote are really intriguing for me as a learner. So naturally, just like you, I started to look for some explanation and I have a guess, though I don't know whether it's plausible. Could it be (I'm addressing this to the native speakers) that 'on a second floor' means in the contexts ' on another floor'
Good idea, but no, definitely not.
and the writer didn't mean to explain to the readers the exact location of the floor in the building? Like 'a couple of minutes' doesn't necessarily mean exactly two minutes. I guess that if the writer had wanted to say definitely what the floor was, they should have used the definite article. If they didn't, then they probably didn't want to do this.
In that case, we'd say something like "another floor" or "a different floor" or "a separate floor."
Just to clarify my thought... It seems to me that in the 1 and 3 examples the meaning of 'second' is 'another', while in the 2 the author wanted to convey the idea that the clinic was not particularly suitable for the patients because they should get over the stairs, but what exactly the floor was was probably unimportant for the writer. That could be the third or even fourth floor... That's only a guess.
And a good one!
I don't know whether it's plausible and would be grateful if the others told me if it is or not.
If they feel that the should have been used there, it's convincingly enough for me personally to follow their advice. Who knows? Maybe the writers of those articles were simply showing off with their creativeness? Wanted something new notwithstanding that it's odd? 
