Yes, that Acurite sensor and the one I have connected to my weather setup (Shenzhen Fine Offset / Ambient Weather) are all AS3935 based. I also have an Aliexpress "JMCU" AS3935 breakout/antenna board that I bought to play with after being pleased with the Ambient Weather sensor, and it seems to work OK to me as well - I had to tune the watchdog thresholds and gain to eliminate false positives, but after doing so it works great.
I did try tuning the watchdog thresholds/gain too but never had much luck. I read a few places that there might have been problems with some of the early AS3935 chips (but could never find too much info) and I know there were problems with some of the breakout boards on Aliexpress having the wrong capacitors on the board (which was another thing I tested/fixed). The one thing I never tried was all my boards are fairly old so they might have whatever the original hardware flaw is and maybe newer ones work as intended.
I've been mighty tempted to try again with a newer one as I really want to build a compact portable lightning sensor linked to Meshtastic but I dread going through the whole ordeal again.
Your trials around the AS3935 piqued my interest a little and I thought that maybe the Blitzortung/LightningMaps people may have tried working with them. It turns out they have! The earlier generation Red board documentation [1] has a section on them and there was apparently internal-to-operators discussion [2] which isn't directly public, but which points to wxforum.net where there were quite a few results with at least one [3] detailing struggles to get it working.
This probably isn't directly helpful but I found it a fun trail to follow for a bit.
[1] https://www.blitzortung.org/Documents/TOA_Blitzortung_RED.pd...
[2] https://forum.blitzortung.org/showthread.php?tid=1022
[3] https://www.wxforum.net/index.php?topic=22235.0