Coin Finish is kind of a catch all term for the gray finish steel.
I'm sure there are lots of definitions on the net all claiming to be the correct one. But as far as making it appear, there are many ways. Here are just a couple.

Some is actually dull nickel or other plating over a frosted/blasted surface. Very good protection of course.

Some is raw steel that has been bead blasted or soda blasted to dull the surface. These raw steel surfaces will rust very quickly in the open air though. Must be protected and a clear coating is advisable

Other ways to get the look are just what you have done,,treat it to a phosphoric acid dip (rust remover).
Turns it a gray tone depending on the steel. You can even up the look with some careful work with common erasers while the stuff is still working on the steel. Works pretty well actually.

A trip through a soln of some brands of toilet bowl cleaner will do it too. Sounds a bit weird, but they have lye and aluminum in it among other things and through all the bubbling and fizzing will impart a nice gray tone to the steel. Tidy Bowl was the #1 recommended product,,for gun finishing as I recall! Never tried it myself..

I've used a very dilute Nitric acid soln (1 tsp/gal water room temp). That works well too. Constant sweeping agitation with a small brush helps even the color as it slowly appears.
Don't try to help it along by heating or strengthening the soln. It will etch quickly and then pit if you're not careful depending on the steel.
It does have a tendency to show up spot hardened areas in some mfg parts w/ a darker color but that can be toned back w/the eraser thing. Worked great on Brownings.

Some call the acid approach passivating (sp?) the steel. Don't know if that's correct, I'm not a chemist and don't pretend to be.
Bluing and browning are of that realm,,a protective oxide layer on the steel.
I'd still keep oil on the surface just like I do with a blued firearm though.

Another way is to put the part(s) in the water while Express rust bluing.
No soln is put on the parts, they just sit in the boiling water while you go about bluing other parts. Once in a while you pull the 'to be gray' parts out and card them to even up the now gray color on them and back into the water they go.
There's enough residue in the water from the rust bluing process as you go along to form a gray color on those parts needing just that.
This method works well, but not everyone rust blues and even fewer Express rust blue.

All sorts of different ways, but unless any of them are a plated finish for the purpose,,I make sure that they get wiped down with oil just like any other blued surface for protection.
a very thin clear coat isn't a bad idea either. Same idea as is put over case color often times.