There are essentially two completely different movements claiming the name of "Messianic Judaism." The first are people who are Jewish- culturally, ethnically, and even religiously, who have converted to Christianity and believe that all other Jews should do the same. There is a small pocket of Messianic Jews of this definition in my hometown, so this is the version I was most familiar with.
It wasn't until later that I learned that there is a second, much more popular movement under the name of Messianic Judaism which are people who are not ethnically or culturally Jewish who have determined that Christianity should return to its Jewish roots. These people have no historical connections to Judaism and usually grew up within a Christian cultural context. There is a lot of overlap with the "Hebrew Roots" movement that you mentioned, and in my opinion there isn't a real distinction between the two.
Myself I feel kind of biased but I view the first kind as more "legitimate" since Judaism, isn't merely a religion, it's a living, breathing culture and it is super weird for someone to just roll up and claim it without having any connection to anyone who was doing it before. It's like if I decided I was going to be Indian and started wearing stereotypical Indian traditional dress and eating only curry because I think that's what Indians eat, without having any actual Indians in my movement.
I agree that ethnic Jews with Christian religious beliefs is a legitimate concept. But I would rather call them Messianic Jews (or just Christian Jews) rather than adherents of "Messianic Judaism." To say that "Judaism" can include Jesus erases the Jewish religion by leaving it without a name, conveniently benefiting the dominant Christian religion. (And Messianic Jews who are not Jews should be called something else entirely.)
It wasn't until later that I learned that there is a second, much more popular movement under the name of Messianic Judaism which are people who are not ethnically or culturally Jewish who have determined that Christianity should return to its Jewish roots. These people have no historical connections to Judaism and usually grew up within a Christian cultural context. There is a lot of overlap with the "Hebrew Roots" movement that you mentioned, and in my opinion there isn't a real distinction between the two.
Myself I feel kind of biased but I view the first kind as more "legitimate" since Judaism, isn't merely a religion, it's a living, breathing culture and it is super weird for someone to just roll up and claim it without having any connection to anyone who was doing it before. It's like if I decided I was going to be Indian and started wearing stereotypical Indian traditional dress and eating only curry because I think that's what Indians eat, without having any actual Indians in my movement.