Heck, I had to stop and take a pen and paper to figure that out. Turns out, you were wrong. (I expected that, but I wasn't sure how specifically.)

As a simple example, imagine that my prior belief is that 0.1 of coins always provide head, 0.1 of coins always provide tails, and 0.8 of coins are fair. So, my prior belief is that 0.2 of coins are biased.

I throw a coin and it's... let's say... head. What are the posterior probabilities? Multiplying the prior probabilities with the likelihood of this outcome, we get 0.1 × 1, 0.8 × 0.5, and 0.1 × 0. Multiplied and normalized, it is 0.2 for the heads-only coin, and 0.8 for the fair coin. -- My posterior belief remains 0.2 for biased coin, only in this case I know how specifically it is biased.

The same will be true for any symetrical prior belief. For example, if I believe that 0.000001 of coins always provide head, 0.000001 of coins always provide tails, 0.0001 of coins provide head in 80% of cases, 0.0001 of coins provide tails in 80% of cases, and the rest are fair coins... again, after one throw my posterior probability of "a biased coin" will remain exactly the same, only the proportions of specific biases will change.

On the other hand, if my prior belief is asymetrical... let's say I believe that 0.1 of coins always provide head, and 0.9 of coins are fair (and there are no always-tails coins)... then yes, a single throw that comes up head will increase my belief that the coin was biased. (Because the outcome of tails would have decreased it.)

(Technically, a Bayesian superintelligence would probably believe that all coins are asymetrical. I mean, they have different pictures on their sides, that can influence the probabilities of the outcomes a little bit. But such a superintelligence would have believed that the coin was biased even before the first throw.)