The second point that the argument that Common Lisp is the "true Lisp" because that's why it was made is a faulty syllogism. Just because that was their goal of creating Common Lisp doesn't necessarily mean that is the case.
And Common Lisp isn't the only language with a standard: Scheme had an IEEE standard less than a year after ANSI Common Lisp, and has several RnRS updates since then.
Long story short, I don't think any language should be considered the "true Lisp". Maybe the original Lisp, but even then only as a technicality. And trying to declare a language as the "true Lisp" comes off as a little insulting to related languages and a little bit arrogant. It implies, to a certain degree, that you look down on other languages as lesser, inferior, failed attempts to reach the true zen of Lisp. I'm sure you didn't mean it that way, but as a fan of Scheme (and Common Lisp too!) it reads that way.
I see your point about arrogance, though I think the insulting tone could be turned around easily -- you have these Lisp-like languages that don't even have Lisp in their name nor a lot of Lisp's crucial features trying to appropriate/claim an inheritance on all the hard work and glory and name recognition that went into Common Lisp and its predecessors. Let them have their name. I'm a pretty big fan of Clojure (and do like what Racket is doing when I read up about it) but imagine trying to sell Clojure to an old Lisp hand with "it's like Lisp" and watching them type (inc "a") vs (1+ "a"). https://pastebin.com/kBfNvxei vs https://pastebin.com/hPB2cy1X and there's no contest. (You might win them over with further persistence but they'll probably be wondering where all their favorite parts about Lisp are in this so-called Lisp-like.)
Does Lisp from 1960 have these things? Then I guess it isn't really Lisp?
Maybe you'd appreciate some Lisp history, to also answer your question about "what other dialect[s]"?
The way I see it, the things that make Lisp "Lisp" are not exclusive to either one. One might say that the intersecting characteristics aren't interesting, but these characteristics are what Lisp is truly about.
SML and Caml are both ML. Haskell 98 and Glasgow Haskell are both Haskell. Dyalog APL and APL2 are both APL. And so on.
Maybe Racket (or Chicken or Chez or Guile or...) has all of these things, Racket is pretty awesome, but those things aren't standardized, whereas I can get those things in Lisp regardless of if I use SBCL/Clozure/clisp/etc. Maybe it's closer to the difference between C (with a bunch of different compiler extensions) and C++ than C and Python, but it's a pretty big difference.
Ultimately I think the claim of Common Lisp being the "true Lisp" is simply that deciding such a thing was the whole point of Common Lisp. Without a standard, you're left with someone writing http://wiki.call-cc.org/eggref/4/multi-methods#examples (or http://wiki.call-cc.org/eggref/4/fast-generic) and someone else writing https://docs.racket-lang.org/multimethod/index.html and neither being able to share code with each other. Most code isn't as trivial as the birthday paradox.
Would you want to try writing a (R6RS) Scheme driver for that 1960 code and see how straightforward it is?
Where do you slot https://en.wikipedia.org/wiki/Dylan_(programming_language) ? Should it be in the running for "the true Lisp"? If not why not?