Is "bench press" a valid name for the bench used for bench pressing?

If the sole purpose of the bench is to do bench presses, and there are other benches in the room, then it seems like the logical alternative would be to refer to it as "the bench press bench". This seems silly, and unecessarily pedantic.

If it was the only bench, then you could just call it "the bench" if you want, although some people might not realise what you're referring to.

So, yes, let people call the bench "the bench press", if it leads to clearer communication, since the purpose of language is to allow clear communication, rather than "satisfy the rules of grammar", which in fact constantly evolve to suit the needs of the population.

"Bench press" as an item (not the exercise) is usually used to refer to a machine for bench pressing. It would include a stack of weights, a bench and a mechanism to transfer your force to the weights. They're less common than they used to be, as chest press machines take up less space.

If you bench press with a barbell or dumbbells, what you use is just a bench. If you're not already talking about a gym context, weight bench or gym bench might be a good way to be clear.

Your case of a bench with a rack attached is interesting. I also use this for other exercises when the other benches are busy, and have never heard anyone refer to it as a bench press in my gym

But if you used bench press, people would almost understand - they'd just think you used a machine rather than free weights for the same exercise.

Just call it the press bench instead of "bench press," that's how I do it.

I don't know if it's a valid proof: but if you put it in google, the suggestions come up:

Press bench -- dimentions/for sale/workout/equipment

vs.

Bench press -- form/record/chart/set