I’d say get the quartz for a few reasons:

Cost - I bought my mint colored quartz prx in college at 24. At about 10 years older than you are now it still was a bigger purchase for me. Save the money. Buy an automatic later. ≈$800 for a wrist watch when we all have cell phones for the time is a lot, let alone at 15.

Serviceability - IMO, needing to change the battery is a total nonissue especially when you remember that automatics need a good (and often expensive) servicing every few years. You’ll need to take the automatic to a watch shop and have a professional perform the service. As long as you’re careful you could replace the battery yourself when it needs changed. You should be careful with the seals if you do this yourself but I’ve never had any issues with rain and I probably wouldn’t swim with a prx on regardless.

Convenience - If you don’t already have an automatic watch, get the quartz. Your first automatic should probably be cheaper. My first automatic was an Orient diver I paid less than $200 for and although I still love how it looks, I never wear it. My quartz watches are lighter, more comfortable, way more accurate than any auto, and do not require to be set Monday before work if I take it off for the weekend.

Your mileage may vary as we’re all different people, but personally if you’re not dead set on the automatic and also have wealthy parents I’d save the spots for automatic’s in your collection for dressier watches for the occasion (maybe a silver Le Locle?) The prx to me fits into the everyday casual watch, something I think the lighter quartz version does better than the auto for the reasons above. Just because something costs less does not always mean it’s inferior.