So i wanna play as close to original as possible, so i'm not interested in fan-made ports and engines. I've installed Quake from gog, and its GLQuake version is 0.95. I faced an issue shown in this video:
https://youtu.be/pXjbgjeKlJ0
And decided to update to newer version, still made by ID. But problem is still there. Any advice?

I'm sorry, but I don't have any advice that you want to hear, but I'd like to point out that although GLQuake was made by ID, it isn't true to the original. It misses overbright lighting and fullbrights that both the original software version and modern ports like QuakeSpasm have. Both are essential for the Quake look. In that regards QuakeSpasm is closer to the orignal than GLQuake!

I'm pretty sure that your problem (the fighting of texture of the elevator and the floor) can't be solved with any version of GLQuake.

Edit: Spelling of the last sentence. Even forgot the "'t". Sorry.

isn't true to the original. It misses overbright lighting and fullbrights that both the original software version and modern ports like QuakeSpasm have.

That's really interesting. Thx.