If you have jumper cables handy, you could properly attach those to the battery terminals, and another convenient 12v battery, and see if the starter suddenly works. If so, good chance the battery theory may be correct.
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while often times this is a safe assumption, and an "easy test", I will say use extreme caution, and be prepared to pull the cables at a seconds notice.....here's why:
I recently watched in terror, what I thought was a woman's engine engulfed in flames outside my office window..... I ran down the hall, grabbing a fire extinguisher, and across the lot to her car where I saw what was happening....luckily.
Seems a pretty non-mechanically inclined sales engineer had attached jumpers to the womans battery, and proceded to hook them up to the battery on his big buck$ Escalade, cranked his car up, and sat there attempting to charge the womans battery, when she hit the key, and began to attempt to start her car that had a "sudden death battery", the jumpers totally melted down, and caught fire, all the insulation was burning/burnt/drripping off of the red hot wires, which began shorting out together while all parties stood there doing nothing.....luckily there was a rag present, and I used it to pull the cables off the Escalade, and then off her cars battery. I almost hosed the dumb guy down with dry Chem just because he stood there watching it all melt down.
She had a typical "hot weather failure where her battery (old) actually overcharged and boiled out, allowing total dead short between all the plates, killing the battery, and when they jumped it....and she hit the key with the other vehical cranking out mega amps....well, let's say they were lucky. I know it had to have shot sparks when he hooked up the cables....
just use care, and always be prepared for the unexpected.
A battery with an internal defect/fault, can explode when an attempt to jump it by attaching to the faulty battery's cables is used.
Another quick thing to check is the condition of the black/yellow ground wire attached to the mounting bolt on the left hand coil, it often becomes damaged at the ring lug, and can refuse to allow proper starting current to latch the solenoid.
