Weight of the Bar?

This is probably a dumb question, but…

When you are lifting, do you count the weight of the bar?

Yes.

20kg/45lb

If you lift an empty bar, does it weigh anything? Yes.

you surely do.

some gyms have lighter bars. 10kg training bars, or 15kg… the women’s olympic lifting bar is 15kg (competition standard).

take into account that different bars do indeed have different weights. My safety squat bar weighs 61.5 pounds.
Some of the big boy bars are 55 pounds if I recall properly and if for some reason you decide in a state of delerium to use a smith machine take into consideration most of them are counterweighted, so they would be 2 maybe 3 pounds at the most !