SOLUTION HERE: I hope this helps some people, as it fixed it for me. I'm not sure if this is exactly the same issue, as there seems to be some discrepancy on this thread.
Quick context backstory. I upgraded to the new Mac mini with M1 chip, with brand new PreSonus StudioLive 16.0.2 USB as hardware/interface. I previously have been using a "frozen in time" version of Logic (10.2.4) with the PreSonus StudioLive Firewire model with no issues for 5+ years. As a creature of habit, making the jump to the current update of Logic on a new computer with a USB board to replace the firewire interface was a BIG DEAL for me!
I monitor everything exclusively through hardware only. Upon loading my standard user project template after finally migrating all my settings, preferences, plugins, presets, and key commands, I tried recording a simple DI keyboard part—audio, not midi—and upon playing it back, noticed an audible delay. The part is behind the beat considerably, and you can SEE IT in the region. It's not a matter of playing back late, it is literally being recorded with some sort of latency.
After some forum digging and troubleshooting, I've isolated the cause of the problem to my master output channel, where my template is automatically loaded with a lengthy mastering chain of plugins in bypass. The problem child is the ADAPTIVE LIMITER. When I try recording with it on, the region waveform is late. When I record with it in bypass, the region waveform is late. When I record after I remove the Adaptive Limiter plugin entirely from the master channel, the region waveform is perfectly in time.
Reference image: Take 7, WITHOUT Adaptive Limiter on master. Take 8, WITH Adaptive Limiter on master.
While a quick fix may be to go about removing all master plugin chains, I realized how annoying that would be if I want to open old projects and re-record or add parts without affecting my working mix & master. This was never an issue in my old version of Logic on my 2015 MacBook Pro, surely it should be possible to keep my templates and workflow the same with a state-of-the-art brand new computer and latest-greatest version of Logic.
More digging and troubleshooting. THE FIX FOR ME: In Preferences>Audio>General, MAKE SURE TO UNCHECK "Software Monitoring" AND "Input monitoring only for the focused track (as in GarageBand)" AND "independent monitoring level for record-enabled channel strips." If you use hardware monitoring like I do, UNCHECK all three of these boxes. For Plug-In Latency, my Compensation is set to "All," "Playback pre-roll" is checked ON, and "Low Latency Mode" is OFF.
I can only theorize that somewhere in the data transfer stream from keyboard to interface to computer, the software monitoring or whatever GarageBand input monitoring thing was causing latency in the signal without accounting or compensating for the lookahead in plugins like Adaptive Limiter. If your workflow is like mine, and you record primarily real instruments and lots of audio tracks with meticulously laid out templates with your own dialed presets, channel strips settings, and plug ins, hopefully the 10 minutes it took me to write out my experience will save you heartache and frustration if you're experiencing weird latency issues upon playback of recorded tracks.