Having stood on my previous cheap white lid to do a job on the extractor fan and fortuitously broken it (the seat not the fan), I have, for the grand sum of £6.98 (less 10% geriatric discount) purchased a wooden "three piece" loo seat, with a completely different, very logically designed fitting that is very easy to install and looks very unlikely to start slipping, coming loose or whatever. I won't even begin to explain it in words. The picture doesn't really help either:
but the "third piece" at the back of the seat covers, and is screwed on to, a sturdy plastic block that is in turn attached to the pan. I hope it works, but for £6.20 you can't really go wrong!