I’m Serbian Orthodox. In so far as we say “All you who have been baptized into Christ have put on Christ,” (Jelici vo Hrista krestistesja, vo Hrista oblekostesja) that would be true. From an Orthodox perspective, that would include all who are immersion baptized three times in the name of the Holy Trinity, and not just Orthodox Christians.
The claims made for Putin seem to me to be what a poorly translated version of what Orthodoxy teaches for ALL baptisms would be: Christ restoring in the individual through His death and resurrection that which we destroyed, allowing us to unite ourselves to God. This is also often used AGAINST baptized politicians who fail to live up to what they are called to through Baptism, a charge I have seen levied against Putin as well.
If the priest is claiming something more, then he’ll probably be dealt with fairly quickly.
Dmitry Tsorionov isn't a priest, just an Orthodox-Nationalist freak. He's not much known in the English-speaking society. In Russia he's got some publicity with several protest actions, which often were of violent nature, and some ultra-conservative statements in social networks.