Presbyterianism is a form of church government where the leadership are required to be elected by the congregation they serve and all congregations are related in accountability to other churches.
As such there are no “independent” churches and also no popes.
But in this the authentic form has three official roles they call offices; Ministers, Elders and Deacons. They think these are pretty well substantiated in the Bible.
Along with that there are other ways of service that are other expressions of the same offices, like the way missionaries, evangelists, teachers of theology in seminaries and perhaps others would be ministers that serve in a different capacity.
For the sake of moral and doctrinal accountability, all of the Pastors (ministers) would also have an official and binding membership in a group they call “the presbytery” (presbyter in the Bible is the Greek word we commonly translate as Elder or Overseer). In other words, every Pastor is under the authority of the other Pastors for the care of their personal, spiritual, practical and doctrinal purity.
The elders that are judges of the congregation, scripture refers to them as “those who rule”, are raised, elected and ordained from the congregation itself. In other words, no elder can govern in a church that is not a member of that particular church. This might seem very obvious to you as a rule of wisdom but it isn’t often practiced. To care for the spiritual well being of people you need to know them and be known by them.
Pastors however have a greater level of responsibility as they are preachers and teachers of the word and so have another level of accountability. Before the term Pastor is laid on somebody they need to have a lot of testing, in the word and in their personal lives. It shouldn’t be easy because the calling isn’t an easy one. The work of the Apostles for example (not to imply that infallibility, inerrancy or divine inspiration are a pastoral norm) would be the work of ministers of the word.
There were elders of course in the Old Testament and the role of New Testament elders is not completely different.
In these days we’ve given over most of the authority of elders that loved and cared for their people over to the civil government. You might not think of your civil servants as elders or overseers ruling over you but what else would it be? A secular government is a church without any higher authority than themselves. This has landed us in an unenviable position of an unlimited, impersonal and arbitrary state that often takes the role of protecting and promoting evil instead of good.
But as long as the state has all of this power what need do the people have of a fully functioning and Biblically balanced church? What the people generally want is a church they can impersonally visit on a Sunday, sing their songs, hear a mild encouragement and get home in time for football. The idea that God has organized the church to be personally vested in their lives is scary to some and annoying to others but that has more to do with choosing the wrong church than with whether it is something God has ordained.
As such, there are no independent Christians, no unaffiliated churches, non-denominational just means a denomination of one church without any accountability, and being alone and autonomous is not something Christ ever gave as an option.
Being in Christ means also being a part of his people, his family in heaven and in the earth.