There are not 2 separate kinds of tongues - a gift and a sign.
The sign of tongues in Mark 16, along with the sign of miracles (such as picking up deadly snakes), driving out demons, and healing people, was a sign to authenticate the early believers and their message in their missionary endevours. Carry on reading the next couple of verses of Mark 16...
Mark 16:20 "Then the disciples went out and preached everywhere, and the Lord worked with them and confirmed his word by the signs that accompanied it."
All those abilities were miraculous powers given to the early believers as confirming signs to others. They would 'wow' others and cause them to recognize that God was indeed with them, and so pay attention to the message they brought. Just as the tongues at Pentecost did. The other tongues in Acts has a similar confirming function - they caused the apostles to recognize that whole new groups of people were also genuine believers and should be accepted into the church.
This ability to perform miraculous signs was given by the Holy Spirit (Rom 15:19, Heb 2:4). It is therefore a gift of the Spirit - the same gift of tongues, miracles and healing as mentioned in 1 Cor 12.
The list of abilities in Mark 16:17 were not given to all believers. Although it happened in the first century, such miracles as picking up deadly snakes and drinking poison without harm are clearly not given to everyone today. Such confirming signs were only for the foundation of the church.
That is the sign giving function of tongues. Tongues spoken in private is not a sign to anyone. In fact private tongues is completely unbiblical. Nowhere in scripture does it advocate speaking in tongues in private. That would be an abuse of a spiritual gift which should only to be used for the benefit of others (1 Cor 12:7, 1 Peter 4:10).
The was also a secondary function of tongues of course when it was spoken in church meetings - to edify others (1 Cor 14) provided it was translated in the assembly.
Charismatics have already invented a new type of tongues, the 'heavenly' tongue to compliment the foreign tongues spoken in Acts which they clearly do not have. Now we have a further subdivision making 4 types of tongues. How many more types of tongues will have to be invented to order to try to get their preconceived ideas and experiences to align with scripture?
A couple more questions..
I take it by this understanding of Mark 16, if there is a sign and a gift of tongues, there must also be 2 types of healing - the sign of healing and the gift of healing and 2 types of miracles the sign of miracles and the gift of miracles. Perhaps you could tell us what the difference between those are?
If Corinthian tongues is the GIFT of tongues, how come 1 Cor 14:2 says tongues were spoken TO God?
If Corinthian tongues is the GIFT of tongues, how comes 1 Cor 14:14 says "if I pray in a tongue"? According to you isn't praying in tongues the SIGN of tongues?
Isn't the 'gift' of tongues in 1 Cor 14:22 also described as a 'sign'?
Sorry but, but like the miracle of hearing at Pentecost theory, there are just too many holes in this idea.