The last name of the president of Russia is Пу́тин. Since people can’t read that without knowing Cyrillic, we need a way to map Cyrillic to the Latin alphabet. However, neither Cyrillic nor Latin script have universal pronunciations: the phonetic value of letters change depending on the language. This leads to the romanization of a name being different depending what the source and target language is. Пу́тин is Putin for Russian-to-English, but Poutine for Russian-to-French. They’re both equally correct, and neither is a change from the other.

So does English, in Russian Putin's name is Путин.

Yep, especially when they come from different alphabets. But we used to do it for English names too (mostly medieval ones though).

Nah, that is actually a slang for sex workers, who do not deserve to be associated with Putin.