As I say in my Comment above I suspect this is a multi-step finish and as such it is impossible to give more that guesses as to how it might have been done. Some experimentation will be required I'm afraid.
At first glance, it looks like some kind of oil over varnish
Oil is never really applied over varnish so I think you can discount that possibility.
In order of strength:
Epoxy will be strongest, but will look plasticky
Oil-based varnish (e.g., polyurethane) is probably second strongest, less plasticky
Lacquer/shellac are probably about equally strong. The benefit is that both are easy to repair. (Though you could repair oil or water varnishes with another coat).
Oil finish (e.g., ...
You could consider another option: don't refinish it at all. Depending on how damaged the existing finish it (you don't say or show) in most case light scuffs and dings are just part of the charm, and don't represent any problem with ability of the material to resist moisture.
That veneer is very stable, with very good natural water resistance. On top of ...
The word you're looking for is "pad", not "pat". What this is referring to is an "applicator pad". This term is not normally used as a verb (i.e. you wouldn't say "padding on poly".) Rather it is a tool used to apply finish.
An applicator pad is a thick cloth or fiber bundle that would be soaked with finish and then wiped over the surface you're applying ...
To me it looks like it is “grey washed”
Instructions can be found here for example: https://www.wikihow.com/Grey-Wash-Wood
And maybe I’m stating the obvious, but remember to first experiment on a few scrap pieces of wood before applying it to your furniture
Thanks a lot to jdv and Graphus for they precious answer / comments. I thought I would post as an answer a description of what I've done, in case somebody faces the same issue.
Take everything apart,
Clean with soapy water,
Apply a mix of 1/2 vinegar, 1/2 water, to the parts that were rusted, let rest 10 minutes and scrub lightly, following this advice,
1/4in dimensioned pieces are going to be a challenge no matter what finish you use, or at all.
As suggested in the comments, you are going to have to choose your wood carefully at that dimension to reduce warping over time. I'd go further and say this is an opportunity not only to look for dimensionally better cuts, but also varieties known for stability ...