Wouldn't the merchant's proof be the two emails they sent regarding the upcoming renewal with charges listed?

Perhaps? Chargebacks are a completely black box though and the entity that makes the decision is the cardholder's issuer so they often rule in favor of the cardholder if there is even a chance they are right (in my experience).

I've had what I thought was bullet-proof evidence before and lost an appeal (as a merchant).

If they sent an email saying or clearly implying it would renew at $38 and the following 2 emails didn't have any clear warning about price changes in the subject line, I think OP would be quite justified in having assumed they were just empty reminders.