"For if I pray in a tongue, my spirit prays, but my mind is unfruitful." (1 Corinthians 14:14)
Paul said that when we pray in tongues, it's our spirits which are praying (through the Holy Spirit), not our minds (because our minds are unfruitful while we pray with our spirits, Paul said). I see "praying in tongues" and "praying with my spirit" and "praying in the Spirit" all mean the same thing.
The Amplified Version of the Bible is often a useful study tool because it brings out various shades of meaning that might otherwise be lost in the translation from Greek to English. Here is this same verse in the Amplified Version:
"For if I pray in an [unknown] tongue, my spirit [by the Holy Spirit within me] prays, but my mind is unproductive - bears no fruit and helps nobody." (1 Corinthians 14:14, AMP)
In other words, when we pray in tongues, the Holy Spirit is providing the words to our spirits which we then speak out of our mouths. This bypasses our minds, which is why Paul said that his mind was unproductive when he prayed with his spirit. ...THis is what happen on the day of Pentecost..the Holy Spirit gave the utterance.
So one type of praying is done with the mind, and another type of praying is done with the spirit. In this verse, Paul specifically said that the way he prayed with his spirit was by praying in tongues in the Holy Spirit. Again, "praying in tongues" and "praying with my spirit" and "praying in the Spirit" all mean the same thing.
Notice that when we talk about prayer, we're talking about speaking to God. Therefore, in this verse Paul was referring to the private form of tongues (i.e. praying to God in the Holy Spirit, or "praying in the Spirit").