Anyone Know What this Connector is Called?

A slight charge could just be a normal battery needs. Wobble takes a little experience and some guessing if its a loose wire, or regulation or normal.

When checking out the old ammeter, see if there is any looseness in that stud.
The other thing to do is check for resistance. It should be so low it doesn't show on a multimeter. Touch the meter's probes together and then check the ammeter. Any measureable resistance is too much.

How the meter works is the metal plate carries the current close to the needle.
The electromagnetic field created by the electricity moving through moves the needle.