I vividly remember the T-1000 killing the dog and the smiling scene when I first saw T2 on TV as a kid, and was always really confused when they weren't on the video. Wasn't until DVD that I realised there was a longer alternate cut with that stuff in. Interesting that they clearly put the Special Edition on TV way back in the 90s, because back then (so far as I can remember) they only ever seemed to air the theatrical release of films on British TV.
For me, T2's a bit like Aliens - the pacing's much better in the theatrical cut, but personally I prefer the extended version and never really watch it any other way. In fact the same goes for basically all of Cameron's films, even Avatar.
Does kinda bug me through that some of the added CyberDyne stuff ruins the neat reference to the first film where the police dispatcher identifies Arnie as the guy who shot up the precinct in the first film. But it's a pretty minor quibble.