Seaquel47 I did a soft shutdown and both the image files and the guiding file were saved properly on the micro sd card.
The two storage types should behave the same way. The process that writes the log files should not care which physical storage it is writing to (something higher up selects). Are you sure that you don't even see the AutoRun logs? (Both AutoRun log files and PHD2 log files are in the same log folder.)
The AutoRun images are in a folder called AutoRun. The log files should be in a folder at the same level called log. Do you at least see a log folder, but it is empty? The hierarchy of folders should look like this:

Just to make sure that the log folder is not mistakenly locked, see if you can delete and expunge that log folder. If you mount the USB stick on a Mac, dragging the log folder to the trash will delete it.
Then remember to do an "Empty Trash" to expunge it, or your log folder will still be in the trash the next time you mount it (with new log files in them!).
Another way is to reformat the USB memory stick and let ASIAIR start with a freshly reformatted drive.
Now try an AutoRun sequence again to see if ASIAIR will create a new log folder (which it should have permission to write to).
I have never seen a case where at least the AutoRun logs were not created. My most recent PHD2 log was from Nov 19, 2022 (third generation ASIAIR). ASIAIR 2.0 public release was, as far as I can tell, Oct 28, 2022, so the log I have (in a Samsing T5 SSD) is from an ASIAIR 2.0 run.

Chen