I have a refurbished X230 that works great aside from this issue. I noticed it would not charge, but when the computer ran out of battery and shut down, I plugged it back in and it *did* charge, that time. I then tried a few things including taking the battery out, which turned the computer off, which then allowed it to charge.

When I plug the AC adapter in, the battery light in the back will blink twice in green, then go back to red (if battery is low) or remain off (if battery level is normal). 

Yesterday night I turned the computer off, plugged it in, turned it on, and saw it was charging. It charged to 97% and then showed up as "plugged in, not charging" this morning. If the computer is charging and I unplug it, then plug it back in, it will not resume charging.

All of this leads me to think it isn't a hardware problem with the charger or battery itself (which holds a charge just fine). I also tried a different charger I had for my old T420 which fit, but had the same issue.

Any advice?

It sounds like you have custom charging thresholds set.  Do you have Lenovo Vantage installed?  (free from Microsoft store)

If yes, make sure you uninstall "Lenovo Settings", and check there under hardware/power.

See below link for Lenovo Vantage.

https://support.lenovo.com/us/en/solutions/HT505081

I use Lenovo Vantage for 2 big reasons.  

1.  Helps me keep software up to date without me having to manually look for them individually.

2.  Allows me to setup battery charging threshold.  I use my computer plugged in to the wall majority of time so i dont like it constantly charging back up to 100%.  Everyones different but I choose to start charging at 60% and stop at 80%.  I find that at 80% if im away from an outlet it still will last me long enough before i need to charge.  Your usage may be different.  I think you said you dont have vantage installed, however if you have lenvo settings installed it has similiar features.  So if its set to charge only to 90% for example.  If you are currently at 93% it will show plugged in, not charging.

Hope this helps.