Based on the comments and reviews, I just bought the Quake Enhanced, expecting to get the original installer with it.
But when I look at my games I only see the offline installer for Enhanced and not original one.
Am I not looking in the right place or was it removed and we only get the new version?

You should have different entries for Quake and Quake Enhanced in your library. The former being the original game you're looking for.

In my case, it's the opposite: I have the original Quake II game, but the Enhanced version didn't came for me. I think that's a matter of few days to appear in my account.

I just checked and the enhanced version appear as Quake II. The old version was renamed to Quake II (Original)
.

For Quake II I have "Quake II" and "Quake II (Original)".
I have "Quake Enhanced" and "Quake The Offering" (installer size 1.1 GB so clearly not the original) but no "Quake" or "Quake (Original)".

Quake The Offering IS the original installer (that was also the name of a compilation pack from 1998). It includes both addons, but it's large because you also get the original CD images of Quake and add-ons (to be able to use CD music).