found it!
and I thought I already tried it, but I must have been too tired yesterday - because I put a mistake in it.
so one of my accessible directories is:
/home/www/USER_ID/html/
my log file resides in
/home/www/USER_ID/html/MYSITE/SUBFOLDER/errors.log
the file triggering the error is
/home/www/USER_ID/html/MYSITE/index.php and all other files triggering errors are next to this file.
The functions for the error handlers are included in a file in
/home/www/USER_ID/html/MYSITE/SUBFOLDER/errorshandlers.php
for the regular error handler I can specify the log file as
$file = SUBFOLDER/errors.log;
but for the fatal error handler I have to say
$file = /home/www/USER_ID/html/MYSITE/SUBFOLDER/errors.log;
Then it'll work! :-)