I don’t think the Chechens thought they were part of Russia. Let’s not forget that the Soviet special forces commander who refused Gorbachev’s order to open fire on civilians in the baltic was a Chechen. Gorbachev fired him for that. He then went back to Chechnya, got himself elected, and promptly seceded. As far as the Chechens were concerned, they weren’t doing anything any different from what the Baltic states were doing. Let’s also remember that the Chechens initially won their independence from Russia. They would still be independent today if they hadn’t foolishly tried to pick off Ingushetia and Daghestan, prompting Putin to re-invade.
Pskov airborne obeyed, but it was farther from Tallinn and it took longer for them to get there. That delay proved very luckily and no one died in Tallinn although it got very tense (In January same year Soviet forces had killed 14 civilians defending Vilnius TV tower). The Tallinn defenders were up in the tower with light weapons and grenades. They had disabled the elevator and threatened that when they can no longer defend the stairs they will turn on the anti-fire system that fills the whole tower with some anti-fire gas and kills everyone inside it. That held the airborne troops back long enough so that the Putchists lost.
This picture is from in front of Tallinn TV tower that day with people blocking the airborne troops.