Engine cranks but does not start - jump start doesnt work
I bought 2012 Fiat 500 a mouth ago, happily. Everything looked good and it drove well. But when I went back home and drove at night, I noticed that the headlights are not coming out. So I went to the mechanic, and it turns out that the dealership sold the car without the fuse in it (that's why the headlights are not coming out), because there was a problem with one of the relays on BCM, causing the headlights never go out. So I had to send BCM to the repair center to fix the relay. In addition, I disconnected the battery while the BCM was away.
I got the BCM back after almost 2 weeks, and when I tried to start my car after connecting BCM, the engine cranked but didn't start. Before this I already knew that it has a bad battery because when I went to the mechanic (to diagnosis the problem with BCM) they said I have a bad one. So I called 2 of my friends and tried jumping my car, but it still didn't start. By the way, the headlights are coming out fine and the EVIC looks good, so I am guessing the battery is not completely dead. Also, the fuel gauge is half full.
Now I'm wondering here - if the jump-start didn't work out, is it not the battery issue then? Even though I get a new battery for my car, the car wouldn't still start? Rolling over your youtube videos, do I have to assume it's either alternator, starter, or fuel pump problem, not a bad battery?
Thank you very much for reading this!