Well, since you pinged me with a link to this...
Not being a native English speaker, I often wonder if I'm using the right word. I like using wordreference.com for this, since it has clear mentions of slang, rude, pejorative, slur, etc, which help.
Now, on to your list...
"Leftie" and "Rightie" sound to me like friendly short-forms. "rightist" sounds weird, and I was surprised to find it in the dictionary. Huh, so it does exist. "leftist" is close to the french term "gauchiste" which is pejorative, so I had written it off, perhaps mistakenly. Your "left leaning politically" suggestion is too long and adds no value, it is just as stereotypical as "leftie". Also there are so many flavors of left & right that one single word is bound to be a broad brush, this is supposed to be obvious. But these words are still useful.
"liberal" ties my brain in a twist since "libéral" means in french "free-market advocate" which is the complete opposite of the english word "liberal" which can also mean anything from classical liberal to marxist (again almost complete polar opposites) depending on who uses it. So the word "liberal" is a can of worms.
For lack of a better option (you can suggest) I'll keep "rightie" and "leftie", first time I used them I put a disclaimer meaning they weren't pejorative. IMO they are not. "leftist" or "rightist" would also work. Maybe someone who actually is a native English speaker could chime in.
"SJW, short for social justice warrior"
It's a thing. Some people identify as such and designate themselves with this term, so the label is valid. I wonder what other alternate term could be used, since it's quite an accurate label for this particular political movement. Maybe "PC authoritarian."
"Cuck, short for cuckold"
Well, implying someone's spouse screws around is more rude than the word itself, I guess. If you refer to the new and rather hilarious use the alt-righters have found for word "cuck" then yes, it's rude and intended as such. I don't remember seeing used here in that way, though. I wouldn't use it, since it's rude.
"Beta and Alpha, short for Beta-male and Alpha-male"
Mmmm, please quote examples. I'm baffled as to why this would be offensive to anyone.
"Nazi, meaning someone is a racist, totalitarian, and/or an oppressor."
If you think the word is offensive, wait until you meet a real one and he invites you to a march. Strangely enough this guy seemed relatively normal before pulling that stunt. I hope I don't need to explain why I never talked to this guy again?
Anyway, I don't see why the word "nazi" itself could be offensive. The ideology is, of course, but it needs a label.
Can we agree that certain terms are offensive and stop using them on the site
Nope, this sounds like a weak attempt to make it more awkward to express stuff you disagree with by tone policing. None of these words are rude, except cuck. None are offensive, including cuck. "Offensive" is a joke about gas chambers, stuff like that.
If you disagree with something, why not use arguments?
So no, "we" don't agree on that.