In essence, the ‘puritanism’ of this selection of authors extends no further than their desire for further reformation of the protestant church within the three kingdoms – hence the identification of ‘puritanism’ with an ecclesiological trend [Crawford Gribben, The Puritan Millennium: Literature and Theology, 1550-1682 (Studies in Christian History and Thought; Eugene, OR: Wipf and Stock, 2008), 8]
What exactly is a "puritan"? In Crawford Gribben's work, the noun/ adjective "puritan" is defined broadly as a desire for further reformation of the protestant church, which historically is situated in the three kingdoms of England, Scotland and Wales. Thus, the word "puritan" can encompass both those who put a higher priority on the unity of the church and thus seek more gradual reforms, even the preservation of things they deem adiaphora (e.g. vestments), and those who put a higher priority on the purity of the church and thus agitate for quicker and stricter reforms. Over time after the Restoration of 1660, those who agitated for quicker and more radical reforms became known as the Nonconformists, who were forced out of the Church of England. Prior to 1660 however, all of them were within the young Reformed Church of England.
The word "puritan" therefore is descriptive more of a mood, or "ecclesiastical trend," than of fixed doctrines or even a certain type of piety. Now, in some modern day Reformed circles, "puritan" is associated with a certain type of inward piety and a striving after holiness. On pastoral practice, we have Richard Baxter's The Reformed Pastor, and John Owen's The Mortification of Sin is perhaps the most well-known work on self-examination and striving after holiness in mortifying the flesh. But historically, this understanding of "puritan" seems skewed. Oliver Cromwell can be considered a "puritan," but I doubt he would be a great example of supposed "puritan piety," at least not unless you are willing to contemplate massacring countless Irish as being a godly behavior. The more episcopal-minded James Ussher can be called a puritan, but a high church puritan doesn't exactly fit the modern Reformed mold of a "puritan," does it?
It is therefore better to use the more historically-based definition of "puritan" as a mood for ecclesiastical reform, instead of focusing on the more inward pietism-lite of a select few. No doubt a concern for inward piety and godliness is part of the 17th century puritan movement, yet its main focus is the reformation of the church. To be a "puritan" is thus to be discontent with the semi-Reformed status of the Church. It says no to the current ecclesiastical situation. It will not settle for the status quo under the partial truth that "there is no perfect church." Yes, there are no perfect churches, but there are churches reformed, and churches unreformed. What we desire is for churches to be always be reformed, going back to the Scriptures and confessing and practicing the historical Christian faith.