There’s a specialist gift of tongues that’s given in church and interpreted, and then there’s the personal prayer language tongue that is a sign that follows all those who believe.
I think you are making a distinction about
the manner in which the
same tongue is performed in the case of interpretation.
It's the same tongue as when the person prays alone, but in this other case, he/she is impressed by the Holy Spirit to utter the tongues out loud in an assembling of the saints.
Then the interpretation comes forth from somebody else, or perhaps the same person who actually spoke tongues out loud.
If you understand the gift of prophecy, interpretation is the gift of prophecy. Really no different, except the manner in which it is utilized. It's an expansion of the interaction of the saints.
Reiterating, with different words -- It is NOT like the person speaking tongues aloud is speaking a foreign language of the earth, and then the person who has the interpretation understands that foreign language and interprets it. That is NOT what happens. The interpreter is getting the prophetic understanding, and is prompted by the Holy Spirit to give the word as a response to the one who spoke in tongues aloud.
For those who do not and have not prayed in tongues, what I just wrote takes the strangeness and mystifying aspect out of it for you. This is why tongues is considered the 1st gift of getting baptized in the Holy Spirit, because you are breaking new ground in the spiritual realm, and you then start to see, via the experience, and get excited about God's power. Most of us express it as though you were getting born again, again. It's a tremendous spiritual experience.
Another analogy: You are a glass of oil partially filled. The oil representing the Holy Spirit within you due to being Born of the Spirit (born again). Now, God wants you to be filled up. So you need to ask like Jesus said in Luke:11:10-13. And take a step of faith to receive it. Then God comes to you with a jug of oil and begins to pour from the jug more oil into you such that the glass begins to overflow its rim. This overflowing is expressed via the 9 gifts of the Holy Spirit (1 Cor 12). But the 1st gift, almost always, is tongues, and extolling God.
It might also include other gifts, but this is seldom seen by the body at large.