I cannot find the Russian phrases for these two stages of dating and am humbly asking for your help. I know that Russians say Маша встречается с Ваней (which literally means Masha regularly meets Vanya), but it is unclear to me as to which of the two stages is implied and what to call the other stage. Also, as a part of my question, how will a Russian man word his proposal to become exclusive to each other? What idiomatic phrase will he use?
UPDATE: My impression from the answer by Anixx and the comment by Shabunc is that many Russians do not really understand this English terminology, so let me try to explain. To put it simply, seeing each other is the stage in which you just go to coffee dates to see people. You are not committed to anyone in any respect in this stage. You just find someone on a dating website, go to a first date, go to a second date, but you are not a couple yet. You just see each other. You cannot become a gf and bf just from the very first moment you see each other. You are not a couple on your first date. It is not about sleeping around or, how Shabunc put it, "polyamorous relationships," but merely about trying to find a good match before really committing to be a true gf or bf. You have to see many people in order to find a good match. You can see different people concurrently. And it is only after you find your true soul mate and after you and that person decide to be a couple that the second stage begins - being exclusive to each other. Just shoot through the following articles to better understand what I am talking about: link1, link2, link3.