What causes T3 timeouts?

That matches up with the answer I found in my own research. Is there anything I can do to determine if the ingress is coming from wiring or equipment on my premises or farther up the line? It would have to be low frequency ingress since error rates on the downstream (high frequency) channels are great. No uncorrectables, and 0 to 3 correctables per day depending on channel.

change splitters and coax. if you have tv service one downstream channel is used for tv, and you wills see uncorrectables on it.

All coax that I can change has been changed along with from the demarc to the pole. Only thing not changed is from demarc to exterior wall jack, but the last tech said that coax was fine even though it's old. No splitters, no TV service.

You'll need to get a tech out that can monitor the power levels in real time.  Your modem will only give you a snapshot.  In the meantime, If possible, I'd simplify the whole path.  For example, I have one piece of direct burial RG6 from the street (via conduit) to my dmarc where it hits a bonded bullet, then from there it's a solid piece of coax all the way to the modem.  I removed all wall jacks and used low voltage cable feedthroughs at the wall gangboxes.  No splitters/taps/wall plates anywhere in the path to my modem (but all I have is internet)

I'm surprised you're not getting any Dynamic Range Window errors along with your T3's though.  Here's my power level snapshot (modem log is clear):  

I had been getting a ton of DRW violations before they changed the amplifier, and maybe I still am. The log has changed in a very strange way where only the most recent event seems to show up at the top, and everything else below it is old (2-3 weeks old). It's like it saved a lot of log entries from the first time I powered it on and doesn't ever write over those. Makes it hard to troubleshoot. Any tips for what I can say to support to make them understand the issue and get the right tech out? Like I said, my signals are amazingly good and I don't think many of them understand there can be a problem even with good signals.

It's strange because it all worked perfectly for about 10 days before the T3s started up. Seems to be at least 2 per hour but it's hard to say since I only get the most recent one in the log. I'm going to try doing a modem refresh even though I'm not sure if that really does anything beyond just a reboot.

The problem with those power levels is like I was saying earlier, it's just a snapshot.  A competent Cox tech can monitor in real time what your power levels are and see if there is fluctuation.  Your issue looks pretty similar to what I was dealing with (good TX-RX power, SNR, etc) and my problem started right after Cox split one of their nodes (Vegas) to ease congestion.  That's when I removed my only spiltter (was trying to keep RX levels closer to 0) and pulled a new cable to the curb from DMARC (3rd time).  Since then, I've been rock solid.

I will say dealing with cox help over the phone or txt is a waste of time.  You need a good tech you can talk to face to face, because it seems you have the vocabulary to troubleshoot this, you just need a tech with more/better tools.