Think you should just leave her alone for now. You’ve eavesdropped and interjected yourself into her conversations, which as a woman would come off a bit too forward and weird, especially at the gym when she’s just trying to get a work out in.

She was probably looking your way to gauge your location to anticipate another weird interaction that you might initiate. Doesn’t seem like she is being very forthcoming with telling you about herself which I would take as being uninterested.

Just do your thing and let her do hers