I'm not confused, and an abacus is a digital computer.

You keep referring to what we are interested in, but that's not a relevant quantity here.

A symbol is a discrete sign that has some sort of symbol table (explicit or not) describing the mapping of the sign to the intended interpretation. An analog computer often directly solves the physical problem (e.g. an ODE) by building a device whose behavior is governed by that ODE. That is, it solves the ODE by just applying the laws of physics directly to the world.

If your claim is that analog computers are symbolic but the same physical process is not merely because we are "interested in" the result then I don't agree. And you'd also be committed to saying proteins are symbolic if we build an analog computer that runs on DNA and proteins. In which case it seems like they become always symbolic if we're always interested in life as computation.

This is where you are confused - in fact just plain wrong:

  A symbol is a discrete sign that has some sort of symbol table (explicit or not) describing the mapping of the sign to the intended interpretation