I've been taking it really easy lately, haven't done any long rides and only ride to and from work, which is local. I'm avoiding racing against traffic lights or trying to go very fast. Would I be better off not riding at all for awhile? And when do I know when I'm ready for more intense rides? I don't want to rush things and end up never healing, or getting worse.

The best answer is to see a physical therapist.
My answer is that it depends on how you feel, honestly. If you feel good doing what you are doing, keep doing it and don't let anyone change that. As cycling doesn't stress the knee joint or its associated muscles throughout their full range of motion, I would incorporate stretches and light full ROM exercises to compliment your cycling. If you feel like you are making it worse, stop. If you don't feel like you are making it worse, there is a huge body of evidence supporting the argument that you are only making it better, faster.
Being ready to incorporate more intensity requires that you and your patellar tendon feel ready to do so - the challenge is to do so gradually. This is where most people **** up, and where a PT can be of much help. Too much too soon and you will be back at square one. A graded re-entry into your previous training is always best given by working with someone who specializes in that sort of thing.

Do you have pain when pressing on the tendon below the kneecap? Patellar tendonitis doesn't seem to be common in cycling.

The pain is on the inside of the knee, beside the patella, slightly above where the knee bends. On my right knee, if I'm facing a clock the pain in in the 9o'clock position in relation to my knee. It doesn't hurt when I poke around with my fingers, only on the downward pedal stroke, when I'm pushing the pedal.

Based on what you planned to do to stop the pain -- guessing that's what had caused it -- you probably have Chondromalacia patella. If so, your plan to discontinue hammering on the pedals in high gear definitely is what you want to do to avoid the problem. From here on out, think of spinning as your friend and pushing as what you want to avoid. If riding a bike at a higher cadence isn't something that feels natural, perhaps consider going to shorter cranks.

Yes, what you have is probably Chondromalacia patella. This is one of the commonest, perhaps the commonest, repeated stress injury (RSI) associated with cycling. It's caused by a muscular imbalance which pulls the patella to one side. This is what I know about fixing it:
https://www.bikeforums.net/training-n...l#post16081392

This stuff works. It's completely fixable. Until it starts to feel better, raise your saddle about 4mm and restrict your cycling to easy rides of about 30 minutes.