These things are easier to understand from an experiential perspective, But...
The "gift" of tongues (prophetic prayer language) that you are referring to is #3 in the list below.
The personal prayer language (#1) would be what was received in Samaria, Ephesus and at the house of Cornelius.
Intercessory prayer language (#2) would be using your personal prayer language to intercede for others. (Rom.8:26-27)
Singing in the Spirit (#4) is from ! Cor.14:15.
The Evangelistic language (#5) you are familiar with from Acts chapter two.
Five Different kinds of tongues
1) Personal prayer language - Speaking to/with God
2) Intercessory prayer language - Praying for others in the Spirit
3) Prophetic prayer language - Addressing the whole church/preferably with interpretation
4) Singing in the Spirit - Singing in tongues/worship activity
5) Evangelistic language - Speaking the message of God to a people in their own language (not yours)
Romans 8:26-27 Amplified Bible (AMP)
Our Victory in Christ
26 In the same way the Spirit [comes to us and] helps us in our weakness. We do not know what prayer to offer or how to offer it as we should, but the Spirit Himself [knows our need and at the right time] intercedes on our behalf with
sighs and groanings too deep for words. 27 And He who searches the hearts knows what the mind of the Spirit is, because the Spirit intercedes [before God] on behalf of [a]God’s people in accordance with God’s will.
Search Robertson's Word Pictures of the New Testament
Romans 8:26
Helpeth our infirmity (sunantilambanetai th asqeneiai hmwn). Present middle indicative of sunantilambanomai, late and striking double compound (Diodorus, LXX, Josephus, frequent in inscriptions, Deissmann, Light, etc., p. 87), to lend a hand together with, at the same time with one. Only twice in N.T., here and Luke 10:40 in Martha's plea for Mary's help. Here beautifully Paul pictures the Holy Spirit taking hold at our side at the very time of our weakness (associative instrumental case) and before too late. How to pray (to ti proseuxwmeqa). Articular clause object of oidamen (we know) and indirect question with the deliberative aorist middle subjunctive proseuxwmeqa, retained in the indirect question. As we ought (kaqo dei). "As it is necessary." How true this is of all of us in our praying. Maketh intercession (uperentugcanei). Present active indicative of late double compound, found only here and in later ecclesiastical writers, but entugcanw occurs in verse Luke 27 (a common verb). It is a picturesque word of rescue by one who "happens on" (entugcanei) one who is in trouble and "in his behalf" (uper) pleads "
with unuttered groanings" (instrumental case) or with "sighs that baffle words" (Denney). This is work of our Helper, the Spirit himself.