In the 30s almost all teams ran a version of either the single wing or "T" formation.
On the single wing the quarterback didn't receive the ball. He lined up as a fullback would today and either the tailback or fullback would get the ball. In this formation you would probably refer to the tailback or fullback as the "back" because they were more important. The slang for quarterback for the single wing is quarter. (There is also the cornerback on the other side of the ball - linebacker wouldn't be referred to as back but would be a backer)
On the "T" formation you have the quarterback, fullback, and two halfbacks (one on each side). In this formation you would probably say I play back as a fullback or runningback.
In general if someone says, "I'm a back." That means they are either a runningback or fullback. If I am running a practice and I say, "Backs go with Coach Shelton!" All of the runningbacks and fullbacks (and wings if you have that) would go. This hasn't really changed much from the 30s. Football terminology is rich on tradition and hasn't changed much.
Now the problem is with the single wing in the 30s quarterbacks were backs, not "quarterbacks". Today no quarterback wants to associate themselves with being a back.
- quarterback = QB (back if in old single wing setup)
- halfback = back
- fullback = back
- runningback = back
- tailback = back
- linebacker = backer
- cornerback = corner
Given your updated edit he is a [full/tail/half]back. Back then those three positions all ran the ball a lot and were for all intents and purposes the same thing. They didn't pass the ball much if at all so you would have 2-4 guys and 3 positions that received a good amount of carries. As a reader I would equate the position to today's Runningback position but know they would have had multiple guys with the same position as that back in the 30s.