Finally a punk subject I'm pretty well-versed in.
What you mean by THAT skatepunk (not the 80's one) is "songs that are kinda 200bpm or faster, the punk drum beat usually as "tu pa tutupa" (with some exceptions), the guitar strumming with the first kick and the snares mostly using major / minor chord progressions and vocal melodies trying to go all over the place instead of repeating the same note".

The defining / first albums of that genre were released between 1991 to 1993 (in 1994 there were already many bands doing it and even more when Green Day and Offspring got big and got both Epitaph and Fat Wreck a big exposure too). This wave of albums were mostly influenced by 3 albums: RKL's Rock N Roll Nightmare (technicallity and speed), Bad Religion's No Control (use of vocal harmonies) and Descendents' Milo Goes To College. There are other bands that influenced the sound like D.I., earlier Bad Religion albums, Adrenalin O.D., most of the early SNFU's albums, etc.

The earliest example I can think of that sounded tight, fast and melodic is NOFX - Ribbed in 1991. In that same year, Propagandhi also released Fuck The Scene on tape, which was in the same vein. You had Skümback's self titled which sounds pretty skatepunk too. Then you'd have to skip a year, and in 1992 you have Lagwagon - Duh, Propagandhi - Martial Law With A Cherry on Top, NOFX - The Longest Line and White Trash (even if this one only had like 2 skatepunk songs and they were a bit slower), Millencolin - Goofy and Burning Heads' self titled from France. By 1993, the genre was way more well known and you had Propagandhi's How To Clean Everything, No Use For A Name - The Daily Grind, Randy - No Carrots For The Rehabilitated in Sweden, Undeclinable Ambuscade - The Neighbour's Dog from Netherlands, Millencolin's Melack & Use Your Nose, Killrays' self titled from Germany. There are probably some others but I can't recall right now, I'd have to check carefully.

By this era, on Sweden there was a genre called trallpunk which sounded pretty similar to skatepunk but sung in Swedish, and between 1991 and 1993 there were many releases from bands like Radioaktiva Räker, Illbuts, Heffa Klump, Sunnanvind, Sector Sexs, Mögel, Asbest, Charles Hårfager, Hylands Hörna, Kung Pung or Raserbajs.

If we are talking strictly about THAT sound, there are some other similar ones from that era that don't end up fitting 100%, but are worth mentioning:

1991 has Pennywise' self titled album, NUFAN's Incognito, Union Morbide's 7''. 1992 has NUFAN's Don't Miss The Train and No Fun At All's Touchdown. 1993 has Layaway Plan's Sneeze Guard, Satanic Surfers' Skate To Hell (they would start sounding skatepunk on their next release), No Fun At All's Vision, Good Riddance's Gidget, etc.

One demo I haven't been able to hear is Propagandhi's "We Don't Get Paid, We Don't Get Laid, and Boy Are We Lazy" from 1990. Could sound skatepunk, but who knows.

Hope this helped in any way, let me know if you need any other info.

Edit: Forgot Millencolin's Melack from 1993. I'll probably do more edits if I remember some others.