A few comments. Firstly, love seeing Happy Birthday To Me and Just Before Dawn as genuinely liked films. Also nice to see Alice, Sweet Alice there, but then, it usually is liked when seen.
Secondly, a second here for House On Sorority Row (the original of course) I think I'd put it in the 'if you like the genre, then you'll probably like it' category.
Pieces
Island of Death (I feel I either liked this one, or hated it)
Psycho II
Roadgames
Maniac Cop
I'd wholeheartedly recommend these (not that the others one the list weren't good) but question whether both Island Of Death or Roadgames are slashers? But I only say that for clarity/so as you don't expect something you don't get, not to have an are they or aren't they arguement because that's awfully tricky with horror, let alone breaking it down into Slasher, with it's giallo/thriller influences. Personally I'd say Roadgames is a thriller and Island Of Death is straight up exploitation, but would recommend both.
I'd also recommend rewatching the original A Nightmre On Elm Street as it's great.
For my own recommendations, a couple of lesser seen entries. The first being Don't Open Till Christmas. It's not good, but it's British, low budget and troubled and, therefore, a bit different to the norm. But I like nasty and seedy and this has both in bucketloads with its atmosphere and onscreen.
A similar feel to my second recommendation (which feels a bit like a first draft of Dressed To Kill, which I'd also recommend, written as a Giallo) again, whether or not it's a slasher is debatable, but as it's made in 1969 it predates 'official' slasher (unless you go back to Psycho/Peeping Tom) but I have no idea how difficult it is to find. It's a kind of Jack The Ripper inspired story which, if you like, I'd also recommend From Hell and the Hammer film, The Hands Of The Ripper.
Lastly I'll throw in The Prowler, which is something you'll have no problem finding and, as you liked My Bloody Valentine, I think you'll like as well.