trinity_gt brings up some good points. Frankly, I have gotten so used to ppl saying "starting" when they mean "cranking" and vice versa, that I failed to fully read and understand the OP.
FWIIW, the battery light is supposed to be on during cranking. The battery light will remain on until the alternator is charging the battery.
FWIIW2, as others have pointed out, there are multiple possible reasons for the symptom experienced. A fuel related issue perhaps being very near the top.
This is why the forums are so powerful. There are an awful lot of very smart ppl reading these posts. Somebody usually knows the answer.
After further consideration, I agree with trinity_gt with the following disclaimer. I had personal experience test driving a Mustang off of a dealer's lot. Multiple attempts were make to start it. The motor cranked what appeared fast enough to start but would not. I suggested a jump. The 1st attempt after the jumper cables were attached, the car cranked and started.
IMO, the odds favor a fuel related problem or failed sensor. Bad battery possible but not likely.