So is Paul saying, "Not a new convert" or "Not a newcomer"?
The quick answer is neither one
This term comes within the context of the qualifications of the office of Bishop. The first thing one must determine is whether there are multiple offices within the Church with multiple titles. I come from an independent Baptist perspective that has one office, with multiple titles for the one office. Roman Catholics or the Episcopal Church allows for multiple offices and a hierarchy among those offices.
I bring this up because it is in looking at some other passages that I think we find the help to clarify what was intended here in 1 Tim. 3:6. To begin with I would contend that versions that translate this term as "new convert" are falling significantly short of what is intended here by this term. KJV has "novice" which is a little better because it their abilities which I think is at the heart of the issue. Yet it too is a little weak on just what is meant. For example, I think Paul hints at this in 2 Tim. 2:1-2
2 Tim. 2:1-2:
1 Thou therefore, my son, be strong in the grace that is in Christ Jesus. 2 And the things that thou hast heard of me among many witnesses, the same commit thou to faithful men, who shall be able to teach others also.
The standard here was that they had to have learned and understood the things that had been taught by Paul. For us this would mean a proper understanding of the New Testament. In addition to having learned the information that was needed they also had to have the ability to teach others. I think it is unfortunate that there are many who fill pulpits that truly lack the ability to teach. In English we have a word that explains some of what is meant here as well as what I think is meant in 1 Tim. 3:6--neophyte. A neophyte was a person who was new to the teachings of their master or rabbi. Eventually they would master the teachings of their rabbi (teacher) and then they would be able to teach others (they would become the rabbi). What Paul is suggesting in 1 Tim. 3:6 and in 2 Tim. 1: 1-2 is that they had been properly trained and now they were no longer a neophyte.
This is why I think "new convert" falls short and why "novice" is a little better. It is not just their age or the fact that they had been a Christian for a long time. They had to have mastered the apostles doctrine to such a degree that they were no longer a neophyte.
An example of age not being the criteria and their abilities in the word is found in Hebrews 5:11-14. The writer in Hebrews 5 had desired to write about Melchizadek.
Then he declares in Hebrews 5:11-14:
11 Of whom we have many things to say, and hard to be uttered, seeing ye are dull of hearing. 12 For when for the time ye ought to be teachers, ye have need that one teach you again which be the first principles of the oracles of God; and are become such as have need of milk, and not of strong meat. 13 For every one that useth milk is unskilful in the word of righteousness: for he is a babe. 14 But strong meat belongeth to them that are of full age, even those who by reason of use have their senses exercised to discern both good and evil.
There the metaphor is of those who are babes and who are unskillful in the word (they are neophytes) even though they should have been teachers at that point. They lacked the ability to teach because they did not have a proper understanding by regular use of the word. I also think it is this last part that leads to the warning in 1 Tim 3:6. A neophyte or a babe lacks the discernment, that leads that person to think more of himself and therefore he is filled with pride if he is placed into such an important role of being a pastor of the church. Certainly thinking less of ourselves comes with age but it especially comes as we are daily exposed to the word of God and we become skillful in the word of righteousness.
I think a similar idea of not just being a new convert but no longer a neophyte in the things of God is also found in John 3:10. In the exchange with Nicodemus Jesus states in 3:10 Σὺ εἶ ὁ διδάσκαλος τοῦ Ἰσραήλ, καὶ ταῦτα οὐ γινώσκεις (Are you not a teacher of Israel, and you do not know these things?). Nicodemus was a member of the Sanhedrin, the leaders of Israel, and yet he does not understand that one must be born again, literally born from above. The KJV translates διδάσκαλος as master because this was used in the relationship between a rabbi(teacher) and his student who was a neophyte. In effect, at that moment Jesus was effectively calling Nicodemus a neophyte even though he ought to have been a teacher who knew what Jesus was talking about.
I think this better explains the term than the idea of a plant, but that is just my opinion.