The definition as taken to the courts in the USA is:
"Hate speech is any form of expression through which speakers intend to vilify, humiliate, or incite hatred against a group or a class of persons on the basis of race, religion, skin color, sexual identity, gender identity, ethnicity, disability, or national origin."
It has more rigorous legal definitions in many other jurisdictions where hate speech is explicitly illegal.
You characterizing toxicity and hate speech as being related isn't a position taken even remotely seriously by anyone who actually write laws on the subject, and many have been written across the world.
Broadly speaking, hate speech isn't "being mean" in any legal definition... But that is what right-wing talking heads like to strawman it as.
I know values and meanings that i have come to understand and relate to those respective words. But you wont find a definite definition online.
For every honest academic attempt there is a bad faith troll. Neither actually embodies a literary authority to enforce a meaning to words.
Language is an emergent construct based on human interactions, all of us that use words are maintaining that fickle construct it in real time. Good and bad actors alike.
If by "a little toxicity" you mean a little bit of aggressiveness, sarcasm, etc., I agree with you. It depends a lot on the community though - in some, allowing it will be counter-productive.
If however you mean harassment and hate speech, as the author of the text, I strongly disagree. If the mod doesn't curb down those things, they might not be "lording" over the discourse, but other users are - because
users shut each other up through harassment
hate speech silences whole groups, as they leave the community
Another detail is that you don't need to control the discourse to curb down harassment, since it's only behavioural and not discursive in nature.
So IMO when it comes to those two things the problem is not overzealous mods, but dumb ones not doing due diligence, who are a bit too eager to falsely accuse their own users to be voicing hate speech or harassing each other when it is not the case.