Bro. Glen, there is an awful lot involved in your question, but I'll try to give a short answer for now. The terminology "Strict" and "Particular" had much more prominent use for Baptists in England than in America. The term "particular" derives from their position on the atonement - that is was
particular or
limited (as opposed to the General Baptists who believed in a "general" atonement). The term "strict" derives from their position on communion - they were strict because they required a profession of faith, baptism, and church membership for participation in the Lord's supper. Most of the English Particular Baptists were also Strict Baptists, although a few, like John Bunyan, were not. For some reason in America the use of the term "particular" was changed to "regular." The early Regular Baptists in America were Particular Baptists. In England, "separatist" most likely refers to those Independents who separated from the Anglican church, but does not necessarily refer to Baptists. In America, the "Separate" Baptists arose during the time of the Great Awakening. Many of the early Separate Baptists were people who left their denominational churches and united with the Baptists. They were similar in practice to the Regular Baptists and later united with them. But it seems for a while the Regular Baptists were skeptical of the
enthusiasm found among the Separates. The difference between the Regulars and Separates was mostly of practice. The Regulars were more formal, more educated, less evangelistic and maybe more pragmatic than the Separates. The Separates had a zeal to recreate New Testament Christianity after having left their stale Protestant homes, and instituted several practices not generally found among the Regulars - feet washing, anointing with oil, right hand of fellowship, holy kiss, laying on of hands, etc.. Some Regulars did practice these things, but not with the consistency of the Separates. Another problem was that the Regulars found fault that the Separates had not adopted the Philadelphia Confession, and that the Separates found fault that the Regulars put too much stock in a man-made document. By around 1800, the Regulars and Separates had laid aside their differences in order to commune and fellowship together.
I don't know that any group of Baptists has ever been called Peculiar Baptists, but a lot of us are "peculiar."
Yes, John Gill was a Particular Baptist.
I guess that to a large extent these must be studied separately, because one could be Particular but not Strict (and perhaps vice versa), and because the Separates had such an impact that they "changed" Baptist history. Their zeal help fill the south with Baptists. But their tolerance of "preaching that Christ tasted death for every man" probably set the stage for the division of Baptists from around 1830 and the years following.