The reason the probability of heads increases is that, once you get a heads on the first toss, you might have picked the biassed coin. Imagine that you flip the chosen coin ten times, and it comes up heads every time. Surely, by now you strongly suspect that you've picked the biassed coin, and you would estimate the probability of heads on the next toss as very, very close to .9. Well, the increase in probability doesn't happen all at once, but increases little by little. Here, I'm considering probability to be a measure of belief.
If I haven't made a calculation error, the probability of heads goes up from $.7$ to $.757$ from the first toss to the second.
The second example seems to be very different. I think you are asking what is the probability that both of them are heads, given that at least one of them is heads, which ought to be more than the a priori probability that both are heads, since we have ruled out the possibility that both are tails. At least, this interpretation gives the $1/3$ probability you mention. Notice that this is very different from the situation where you flip two coins and randomly choose one of them to look at. If it turns out to be heads that tells you nothing about the other coin. It's more like a situation where you toss the coins, and I inspect them and tell you that at least one of them is heads. If you are to bet that both are heads, what are fair odds? (If it turns out that I lied and both are tails, you win.)
If we perform this second experiment with the original two coins, we still find that the probability increases. That is, if we toss the fair coin and the biassed coin simultaneously, the probability of two heads is $.45.$ If we are told that one came up heads, the probability that both came up heads rises to $.45/.95\approx.4736.$