The criteria is pretty straightforward: you apply the "intolerance of intolerance" to the first intolerant action in the chain, i.e. the one that infringes unprompted on someone else.
So, the gay couple coming to the baker that doesn't want to make a cake for a gay wedding are the intolerant ones because they are the first intolerant action in the chain? Because you know that will be how it will be spun.
Congratulations on the apt handle. Your argument is irrational because it ignores the fact that the baker committed the first intolerant act in the chain, not the gay couple.
It isn't irrational because every human being involved experiences a different intolerant act as their first intolerant act. The baker experiences the customer as committing the first intolerance to him. The customer experiences the baker as committing the first intolerance to him.
There is no global ordering of intolerant events. This is basically the CAP problem. Every node (or human) sees their own version of the database (or the world) unless all humans coordinate with all other humans (aka impossible).
So refusing to provide service is "intolerant"? Does this only works on "protected groups", or does it work in general? How about the baker is willing to provide service to the gay couple, but just not the gay wedding? Now, please elaborate how you define "intolerant".
Absolutely: refusing to provide service to gays, when you provide the same services to other people just because they are not gay, is indeed literally intolerant, without your scare-quotes or any other qualification.
What's so hard for you to understand about that?
If you're trying to argue for some slippery weaselly nuanced non-standard definition of "intolerant" which excludes bigoted bakers that you just pulled out of your butt, remember that it's a double edged sword that cuts both ways, and also excuses gay couples for not tolerating homophobic bakers.
It's not my responsibility to provide you with the standard definitions of common English words, when you're obviously capable of googling them yourself, and obviously misunderstanding them on purpose, and obviously not arguing in good faith. Look it up on Wikipedia yourself.
The baker in your chosen example is very convenient in that they are clearly anti-gay. Consider the the real life examples of bakers who are allegedly happy to serve gay couples, but believe that baking a cake for a gay couple's wedding would be a speech act in which they do not wish to engage, or a hotel providing 'separate but equal' treatment to people of colour. These things strike me as problematic, but clearly were not obviously so to the legal system of the time.
I think you're likely to run into the general issue that people seldom phrase their motives so as to make themselves sound unreasonable or intolerant.
The baker's argument was that it wasn't the same service. They would have baked them a cake; but they didn't cakes with "jim and john's wedding" written on them, in the same way you wouldn't bake a cake with the 14 words on it, even if you'd bake a cake for Richard Spencer.
I don't bake cakes for Nazis, no matter how many words they want on it. Simple as that. Not even cupcakes. No nuances.
It's pretty obvious when the baker and their supporters start bending over backwards to make nuanced hypothetical situations and ridiculous unbelievable qualifications, that they aren't making good faith arguments. If their best and most honest argument is that their bible told them to be intolerant bigots, then that's their problem for choosing to take their marching orders from that particular bible, while choosing to do business in that particular state which bans discrimination. The fact that your bible tells you to do something illegal is certainly no excuse for stoning your wife to death or killing gays, either.
So we should have the conversation in which everyone has to make the best arguments they can, instead of trying to go recursively meta with the Paradox of Tolerance, accusing the gays of being intolerant of the baker's intolerance. Simply judge them all on the merits of their best arguments and intellectual honesty and willingness to address valid counter-arguments.
Is this confounded or bound up with things like fake news or bad science -- would I, if I owned a platform – call it π-Chan –be prohibited from banning users who espoused anti-vax ideas because I firmly believed they put society at risk (or whatever the justification may be)?
I would, on the one hand, be strongly supported by science, but on the other intolerant of others' who have differing points of view. Would I be allowed to run my platform the way I wanted? Whose rights would prevail in that case?
The question is "who do you want to be your consumers" here. Both choices are PR choices. Legally, you're allowed to do either ("neutral platform" and "moderated discourse forum" are both things that are legally allowed to exist in the US at least).
If you want to appease the free speech crowd, you let the anti-vaxers stay (Gratz, you're running a Chan!) If you want to appease the intelligent discourse crowd, you ban/moderate people who violate your TOS (Gratz, you're running something closer to Hacker News!)
And if you can´t see why the above applies, maybe you ought not to use the word "straightforward" anymore because you are clearly not qualified to do so.
Then you have a discussion about whether that is right or wrong, instead of having a discussion about how recursively meta you can go with "intolerance of intolerance of intolerance of intolerance of intolerance of ...".
Call it the "no recursion" rule.
Case in point:
The baker who refuses to bake a wedding cake for a gay couple is intolerant of gays. That is unjustified and wrong.
The gay couple who sues the baker who refuses to bake a wedding cake for a gay couple is intolerant of intolerance. That is justified and right.
The baker is unjustified and wrong to complain about the gay couple's intolerance of his intolerance, because he was unjustifiably intolerant himself, first.
Ok, let me be facetious. Take the baker situation and push it to full overdrive.
Let's say the baker was a victim of sexual abuse and the client was asking to bake a cake rape-themed. (Not in support, you can suppose it was a replica for a documentary let's say). The baker refuses, the client sues, who is the first intolerant?
Another case, a client enter the shop of a muslim baker and asks for a depiction of Allah on a cake. Who is the first intolerant?
Maybe you have answer for all such situations, but how confident are you that a majority agrees in all cases?
I think both these cases are fundamentally different -- the baker can refuse to bake a certain type of cake, on the grounds of religious expression clashing with the cake he is making. However, it's different if he refuses service on the grounds of the type of people he is making the cake for.
and in the actual case the baker refused to write a message on the cake. he was fine with the clients and with the cake.
He only objected to a custom message defying his belief. There are good arguments on both sides. (my stance is sort of about scale as in if he had enough employees to delegate or he was the only cake-maker there)