You're right, it's not a false widow. I'd say it's a lace weaver spider, probably Amaurobius similis. That dark mark at the front end of the abdomen, surrounded by the lighter colour is a clue. You can't see the segment that the legs are attached to (cephalothorax) in this picture but it would be darker at the head end end if I'm right.
It's a common spider that often tucks itself away in crack, holes, spaces under bits and pieces in sheds etc. and leaves a lacework of silk around the entrance to its retreat. You're by no means the first to ask about this spider on this forum; it is one of several which get mistaken for false widows: