I had already thought about storing errors in the table / separate file / logfile and it seems to me that the best way is to output errors in logfile. As for user it is more convenient to look for errors in the place where they are usually generated - in logfile and if he wants to intercept them he could easily do that by few commands.
The analogues of this feature in other DBSM usually had additional files for storing errors, but their features had too many options (see attached files).
I also think that the best way is to simplify this feature for the first version and don't use redundant adjustments such as additional files and other options.