I thought someone told me you never use more than 2 coats of primer, but maybe I'm doing something wrong, because I can still see the dark gray paint samples we put on the wall, I don't have a nice bright white wall.
Was I being too easy with the primer and should have applied it more heavily? Do I just need to primer over the whole wall again, and again, until I can't see the old paint samples?

 

No need to add a coat of primer.
And there was no need to even use that second coat of primer.
Just going to need two coats of the finish paint.

 

Some primers cover better than others. Different primers have different tasks so it's possible that particular primer wasn't well suited for the job. Application error is also a possibility. Technically painted walls don't need a primer if being repainted by the same type of paint [latex over latex] but it can make it easier to get coverage by the top coat. Some brands/lines of finish paint cover better than others.

As Joe suggested, I'd proceed with the finish paint. Depending on the color and quality of the paint it should cover in 1-2 coats. Don't try to squeeze all the paint out of the roller while you are applying the paint. I like to apply a stripe of heavy paint and then take the roller cover [now with less paint] and reroll the previous stripe to level it off.