I'm unsure if this is better than the original sentence. "tokens" is something that the reader at this point doesn't understand and the read flow looks better when starting with "No, ...".

I don't agree on this one.API token is used a few line above, and guards are the main entry point now. Indeed, the flow is broken but I propose to change it all together. The anonymous concept has always been confusing and deserves to be clear as soon as we can since it is the first setting to start with, the sentence itself is clear enough IMO.
Furthermore, before the next line we encourage to see the WDT showing an authenticated anonymous user with an anonymous token, currently we're saying that Symfony is tricking us, instead of being explicit without explaining all the internals. Enough words :), what do you think of my new change?