Some of the words that fit your objective are mortise and recess. Typically a mortise is an opening cut part way into wood to receive a tenon to create a strong joint. A mortise is described also as a recess. You can also use inlay to find other useful references.
For a large area project as you describe, a set of chisels and gouges would be useful if you ...
What is the difference [between glue drying and curing time]
As covered somewhat in this previous Q&A, How long does glue take to dry? all PVA wood glues 'dry' by literally drying initially and then there's a curing phase, where they undergo a chemical/structural change and harden more completely and get closer to their full strength.
So your glue says ...
From the Titebond website:
What is the clamping and drying time of Titebond Wood Glues?
For most of our wood glues, we recommend clamping an unstressed joint
for thirty minutes to an hour. Stressed joints need to be clamped for
24 hours. We recommend not stressing the new joint for at least 24
hours. For Titebond Polyurethane Glue, we recommend ...
Drying is what you get with brine. The water dries, leaving the salt. It's still salt, just not wet now.
Curing is what you get with epoxy. The parts turn into something else entirely.
All paints do both. Curing is what makes them not wash away at the first exposure to their own thinner.
They do not do both at the same rate. That's why you can have ...