As far as...
The physical Israelites, James 1:1, and Matthew 13:52 were both.
goes as being both old covenant and new covenant. I haven't had anyone come so far as to be able to explore this space.
What does being in the old covenant give a person? What does being in the new covenant give a person?
Is it a conflict of interest — i.e. can you be employed by two covenant contracts at once?
Could those of the old covenant even be of the old covenant anymore...?
I am curious as to your quoting James 1:1. I love the letters of the apostles to the dispersion (they, for instance, point out that they knew the twelve tribes existed in dispersion, though only two tribes were in the land prior to 70CE). But not sure how it shows that they were in both...?
Trying to reason out loud—and forgive me I'm in new ground as I have only conceptually daydreamed not critically thought about this yet, I was looking for you guys to do it with first! :)
—I think that after 70CE there was no longer the ability to participate (or live fulfilling the obligations of the old covenant). It was clever the way that all went down, so that while faithfully people of the old covenant today do as far as it is up to them to keep the commandments, they still cannot offer sacrifices in Jerusalem, and other temple/priest focused things. That ended by 70CE.
So I guess I don't think of those in the dispersion as "both" but rather part of the "new". If you "over simplify" you could say that to go to heaven (new cov) would mean accepting Christ as Messiah, his sacrifice as having fulfilled law, thus releasing you from slavery to the law. You could either accept it and move into the new covenant, or not and deny Christ.
The COMMANDMENTS must still be kept as noted in another thread, but that's just the commandments, not the law. Law including circumciscion (there wasn't a 10 commandment for circumcision) and sacrifice (also not one of the 10 commandments). There were others, but those two make the point of the difference between keeping the Law (as the section of scripture as designated and referred to by Christ ( Romans 3:21,Matthew 22:40, Matthew 5:17,Acts 13:15,Acts 24:14,Zechariah 7:12,Jeremiah 2:8 ) and the 10 commandments which happen to have been written inside the (books of the) Law.
So I conclude that I don't think you can be a part of both. Simplest: You can be apart of the new covenant without recognizing that the old covenant is no longer valid. It's time, in and by Christ, had come, therefore it gave way by his fulfillment in sacrifice, and opened the way to the new covenant.
For those in the old covenant who do NOT recognize the old covenant (there are new Jews every day!) then there is a satisfying completion to the story for them... but that's how it goes—different rewards, different ending, until one day we are all together again in one family!
You asked about if I have done research on the covenant types, yes, but I haven't discussed it. I researched more individually rather than as "types". I'm aware of only the two types, and don't have a list of them yet.