Interesting, although on one level that doesn't surprise me; (that you've never encountered cessationist teaching). You don't appear to be too familiar with the Reformation either. Which is rather interesting (that you don't know the history of the Reformation) seeing how you state you are an ordained Presbyterian elder. (Presbyterianism came out of the Reformation and all the reformers were cessationist.)
The Biblical definition of speaking in tongues is shown in Acts 2. Clearly the apostles were speaking in what was being received as decipherable foreign languages. Thus assuming upon the consistency of Scripture; (seeing how we are instructed to build doctrine upon comparing Scripture with Scripture) that the tongues spoken in Corinth were consistent with what happened in Acts. Which clearly, were received as foreign languages.
Many scientific studies have clearly proven that glossolalia is not a language. Multiple religions around the world practice glossolalia. It's not a miraculous phenomena. The tongues practiced by modern pentecostals has also been proven by scientific study to be glossolalia; and therefor is not an actual language.
"Speaking in tongues" did not become a thing in the Christian church until starting about 150 years ago and didn't go "mainstream" until after WWII. This is not a practice that has existed for centuries. Why do you suppose that is?
It is interesting that Calvin said in his commentary on 1 Corinthians 14, that tongues and prophecy declined because of misuse and that these gifts ended up being discredited and so believers of the time did not seek for them so that they were not identified with the misuse. He definitely did not say that they declined and ceased through some decree of God that they cease after the Apostolic age. I studied the Reformation while doing my MDiv, and I am not sure whether Luther was cessationist or not. I think he did report that some got a bit ethusiastic and spoke some unusual languages, but don't quote me on that. John Wesley reported that he heard of some who "gobbled like geese", which is not surprising because I have heard some speech in tongues that sounded just like that.
There have been movements over the centuries that have been very charismatic in nature, and because the only record we have of them is through the Roman Catholic Church's court transcripts, so we might have to read between the lines to determine whether they were true believers or not. Because the RCC were enemies to any movements that would not acknowledge the Pope, then I wouldn't trust any conclusions the RCC would have concerning them. Most of the literature of these movements were destroyed so we don't really know what they were about and whether they displayed charismatic gifts or not.
What we can't discount are the many testimonies of people in Pentecostal churches speaking what sounded like "Glossalalia" (or gibberish as you say), and yet people in those gatherings have heard their own native languages being spoken. I had a friend in a prayer meeting at my church who spoke in tongues and a Ghanaian visitor heard him praising God in his own rural village dialect, and I know that my friend was hardly able to express himself in English let alone an African rural village dialect!
I have often travelled on the bus to work with people speaking Hindi and Mandarin Chinese, and both languages sounded like gibberish to me, and yet they understood each other perfectly. Just because a language sounds like gibberish doesn't mean that it actually is.
Many cessationists have a blind spot when it comes to tongues which causes them to discount these many testimonies of people speaking in tongues and being understood. I think it was St Francis of Assisi who had the experience of needing to share the Gospel with a Frenchman and after asking the Lord for help, started speaking in a language by faith and found that the Frenchman replied in the same language. St Francis discovered that he was speaking in French to the person, a language he had never learned. There are many testimonies especially during the early days of the Pentecostal movement, where people who had never learned a language, actually spoke it and was understood. There is the testimony of a woman who prayed for the Italian language so she could share the Gospel with those in the Italian section of her city, and was given the language and was able to speak it for the rest of her life.
These are testimonies that I have read about in my extensive reading over the years, and it would be no point trying to actually cite them in the literature; anyway, you probably wouldn't believe if I was able to.
I know the stories about some early Pentecostals thinking they could go to India, China and Japan and give the Gospel in tongues, and came back home disappointed and disillusioned because it didn't work for them. There are over enthusiastic people in all religious movements to do things out of presumption instead of faith.
