Start here.
That's a huge red flag/clue. Unless you've got a Chris Bryne gorilla grip you not be able to wiggle it off if it's set properly. Clean to shiney lead, reset and seal terminals properly.The 'live' cable from the battery is a little loose... its fully tightened but if you wiggle and jerk it, it will pop off the terminal.
An amazing amount of volt drop occurs right there in a dull loose connection.
Charge and test the battery on a load tester. Then and only then with clean tight terminals start thinking about other possibilities.
Every solenoid, starter and cable possibility is a red herring until this is put to bed and confirmed good.