Is the Day One patch included in the download from the site or is it separate?

This is just an educated guess but the game version on gog, browser and galaxy says v1.01. So I think it's the release version. There are no patch notes available on gog, neither are there any on Steam. I am not even sure if there is a day one patch. The only thing that was patched for sure was the Steam Early Access Version, which wasn't available on gog. There is a Release FAQ on Steam which states that the EA Version will be patched to the release version.

But I am pretty sure there will be some patches coming our way soon-ish, which is pretty much standard nowadays, and gog will release these updates as fast as they can.

The PS4 and Xbox versions also got the Steam patch, the change log is here:
https://www.reddit.com/r/Torment/comments/5w15kk/day_1_console_patch_notes_8_gb/

I'm sure there will be more coming, but this one looks substantial so I didn't want to start on the earlier version, but if this is v1.0.1. then it is the patched version.