I have just contacted their support for the second time. Let’s see.
To sum up so far: Firstly, I received that email from Payza saying that my transaction was completed; then, after many days seeing that my transaction was still unconfirmed, I contacted the support for the first time, but didn’t get a solution. My hope was that the bitcoin, after weeks unconfirmed, would eventually 'drop' into my wallet. It didn’t happen, Payza didn't return the money to my account, and I am at a loss now.