Is this a starter problem?

2011 CRV. It has a weak start, and cannot start twice in a row. When jump started however, it starts strong. So I figured battery.

But I put in a brand new costco battery, and its back to starting weak again.

I have not load tested either batteries, only checked voltages. 12.6 old, 12.8 new.

Is it possible for a starter to just be picky about voltage? Like it only likes the higher (13+) voltage from another running engine/alternator/battery?