The vibration is unusual, but we’ll forget about that for now.
The difference means it’s at least noticing that the battery is inserted and it’s probably trying to charge it. That doesn’t tell us where the problem is, but it’s more hopeful than if your phone didn’t detect the battery.
Best case scenario is that your battery isn’t making good contact. Loose batteries are a fairly common issue in FP2s of a certain age.
Can you try to insert a battery shim to squeeze the battery tighter against the gold contacts? Just a bit of cardboard, tape or a bit of folded up paper will do, at least to see if it does anything.
It’s also a good idea to clean the battery contacts with some isopropyl alcohol or contact cleaner on a cotton bud.