Ukraine absolutely, and desperately wants NATO involvement, if nothing else to strengthen their hand at the conference table. I am more skeptical about the Polish, Czech and Slovenian governments. Poland has every reason to want the strongest possible NATO commitment to defend NATO territory, and to escalate short of WMD exchange as a consequence. That's the hand that they've been playing up until now, though I don't know how committed they are to that approach. The Czechs and Slovenians have a better dirt situation, and I have a hard time seeing them as provocateurs. Someone like
@Bruce Wayne might be able to weigh in on the situation on the ground in Czechia, and prove me wrong at least in terms of the immediate politics.
He didn't end up with a long-term separatist conflict in Georgia, though. He forced Georgia to accede to letting go of his sympathizers and walked. He more easily could have taken much of Georgia if he really wanted. I wouldn't have wanted to pacify an insurgency there either, though. Trying to dig insurgents out of the Caucasus is just asking for trouble. See: eastern Afghanistan and its own mountain range that the British, Soviets and United States all broke their teeth on.
In the case of Ukraine, the dirt is more favorable from an attacker's perspective but they have 10x the people. This necessitates a lot more boots to pacify an insurgency, and other posters have highlighted some of the evidence from over the years that despite internal divisions, the Ukrainians respond strongly to threats to their identity and circle up the wagons when those threats occur. Sure, some of that evidence might be a clip from
Seinfeld, but the clip is believable (and funny), which is the point. If the clip weren't believable, it would tell us something about who they are.
What Putin probably wants in Ukrainian politics is influence, more than the dirt. Reality is that Ukraine has made a decent chunk of Russian natural gas exports from pipelines crossing their territory vanish over the years. The dirt would also be as good as it gets from the perspective of NATO launching disabling counterforce missile strikes against his land-based forces. From his perspective, these are the threats.
The narrative seems to be that Putin went into Ukraine whole-hog because of the cultural linkages, and thought that he would be welcomed with open arms, but I think that's wrong. I think he went in as hard as he did because of the stakes. I don't think he expected to be welcomed, but I also think he believed he could project more hard power than it turned out he could. I think he expected to roll up the dirt east of the Dnieper fairly easily, and produce a failed state incapable of joining NATO while driving a hard bargain for peace.
You are, of course, welcome to dispute the non-factual portions of that argument. It's my assessment. It doesn't mean I'm right. I'm no expert on Ukrainian or Russian politics. I know an awful lot about the system more broadly, but I won't presume to claim area specialty. To use a poker analogy, I can put Putin on a range based on the available information. I can't necessarily put him on a hand.