I like the original post you are referring to. Concerning that post, I would say that (as per the annotations of the Bible I use) 1 Corinthians 14:2 does not refer to a language we speak but the meaning behind what we say. For instance, I could say something as simple as The Truth is a living being. And people would not understand its meaning, but only those that have learned from God to understand it, for it is a mystery. (The Bible I use also has a secondary interpretation for this verse, by saying that some scholars belief he was referring to Hebrew as it was not a widely used language).
For the sake of another example, we hear a lot “God is love”, even in atheist people, who would use this sentence in their discussions. But we know they do not understand this language; and we believe we do, for we can tell we understand it beyond infinity, by the grace of God. However, the truth is that even for infinity it is an unknown language for us, as it is a mystery spoken by the Spirit. And indeed when I say “God is love” my heart and my mind grow, as if God would teach me lessons just by saying it or thinking about it, but it doesn’t mean it will necessary teach other people, unless they repeat this language in their mind and heart (1 Corinthians 14:4).
Even if there is someone to interpret me saying “God is love” (a believer), it doesn’t mean this person could repeat it so that some else could understand it using the same gift of interpretation, which makes him/her a believer. This third person would be a using the gift of a Prophet when helping other people understand more this language.

Your explanation of the way you use the phrase “Tongues of men” is good. But I refer to “tongues of men” using the name “Gift of Tongues.” What you call gift of tongues, I do not believe it is a gift. So this part of the topic gets to a dead end for you and me; i.e. I would not discuss it further.

As for the language of Angels, which is in 1 Cor 13:1, the Bible I use has a note that reads: “Even if I were to speak the tongues of all the nations of the word, or if I were to make myself be understood without using my voice or an external sign, as the Angeles do, which not needing to speak, communicate immediately their concepts, which would be even more praiseworthy than speaking all the tongues of the word, etc.” (a fast translation made by me).