I just started learning to play tenor saxophone which is tuned in Bb key. All the tutorials begin wit learning the C major scale, but this is actually the Bb major on a tenor sax. I sort of understand the reasons why there are transposing instruments (answered in this question), but I'm still confused about under what names should I memorize the tones and scales I'm practicing:

1. Should I learn the tone names relatively to the instrument tuning, e.g. when I play Bb on a tenor sax or trumpet, should I call it C?
2. Should I learn the absolute tone names e.g. when I play Bb on a tenor sax or trumpet, should I call it Bb?
3. Should I learn both?

The first approach seems to be the way playing on a transposing instrument is being taught. It will apparently make reading sheet music and switching between differently tuned instruments easier, but on the other hand it seems to bring a mess into communication with players of other instruments, which wouldn't be an issue wit the second approach.

I have encountered this issue many years ago when I was accompanying my cousin who played on a trumpet with a piano. I told him to play C and he played Bb. When I told him that it is not a C, he could not believe it. He was already playing trumpet for maybe two years at that time and his tutor was one of the greatest jazz trumpet players ins Slovakia.