Is the English translation accurate? I would expect “Er wagt es nicht, etwas zu sagen”. The German sentence looks like “He doesn’t trust himself to say anything”, which doesn’t feel like it has the same meaning to me.
“trauen” and “vertrauen” are not so different. So it’s not surprising to get them mixed up. It’s only the reflexive version of “trauen” that also has a different meaning:
sich (acc) trauen - to dare
sich (dat) trauen - to trust oneself
sich (dat) vertrauen - to trust oneself
Ich traue mich - I dare
Ich traue mir - I trust myself
Ich vertraue mir - I trust myself
So it’s only with “sich” that it becomes ambiguous, because “sich” is the same for dative and accusative. But then, it’s uncommon to say that you trust yourself, so the default interpretation is “to dare” (when it’s not unambiguous because you can tell the case from the pronoun).
But I actually think the “different meaning” is not so different after all. I’d say if you substitute “to trust” with “to have faith”, you can see that it’s not that big of a step: I trust myself → I have faith in myself → I dare.
Now, for the difference between “trauen” and “vertrauen”… It’s actually not so easy.
I’d say “trauen” is more situational, immediate, and concerning one single thing. For example you trust that some statement is true, of if you don’t trust, you suspect a scam, etc. You can trust that a layer of ice holds your weight. There’s a phrase “Ich traue dem Frieden nicht”, which basically means “something’s fishy”.
“vertrauen” is more general and long-term. It expresses familiarity, and means you are willing to give up control, your money, your life, etc. and put them into the hands of the one you trust. Which also means it is usually only used with people, because you believe in their good intentions, which inanimate objects don’t have.
Ich traue dem Eis. (It will hold my weight and not break) Ich vertraue dem Eis. (sounds weird, unless you want to anthropomorphize the ice)
Ich vertraue darauf, dass das Eis hält. (but this works )
I think english makes a similar distinction with “to trust” vs. “to trust in”?