Alex,
Yes, exactly. Alternatively, as ersa suggests, add another column into which US safety pilot time is recorded and alter the [EASA] PIC time accordingly. If you do this then strike out the old PIC times using black ink so that the original entry is still legible; correcting fluid is not viewed favourably in the North America.
... error due to bad advice, better to let it show than have to explain a logbook with new pristine entries.
May I accuse you of logbook bigotry?
It is not an error to log US training in accordance with US rules.