Thanks for listing the ideas of what constitutes a continuing Christian church. As far as the LDS church is concerned, and I'm fairly certain the Catholic church as well, it's only one thing. It's the "one aspect" of priesthood authority. Whether it is through apostolic succession, or that it was restored in the latter days directly from the original Apostle(s) during visitations.
No other restored church outside of the LDS claims to have had the priesthood restored in such a fashion.
While the LDS church declares that priesthood authority is necessary for valid baptism, the Catholic church accepts baptisms done outside of the church as long as they were done with a Trinitarian formula (in an emergency, they can even be performed by non-Christian as long as they are done "in the name of the Father, and the Son, and the Holy Spirit). Not being an expert on Catholicism (I'm learning), I suppose Catholics believe that the priesthood is necessary for the various rites and to offer the Eucharist.
As an aside and to tie this back into the original thread topic for minute, it does seem like Catholic priests do have more power readily available to perform exorcisms than say Evangelical preachers. But that is another topic.
Anyway, you're correct, it does matter what the definition of a continuing church is. Perhaps, that is what I was wanting you to tell me.
I'm open to opinions. Yet I tend to see the necessity of an unbroken chain of authority... Thanks!
As a side-note, Presbyterians -- even the more Conservative, fundamentalist-Calvinist sorts -- also accept Roman Catholic baptisms as Valid, due to the Trinitarian nature of the baptism (as well as accepting the validity of Eastern Orthodox and other Trinitarian Protestant baptisms). So, I guess I'd say that I think that part of the definition of a "continuing" Church has to be a Church or group of Churches which have consistently maintained the Athanasian doctrine of the Trinity since Apostolic times. Because that does seem to be a consistent theme of "universally essential doctrine" amongst Roman Catholic, Eastern Orthodox, and old-school Reformed Protestants.
Just ruminating on the definition of what constitutes a "continuing" Church; I'll give more thought to the rest of your reply tomorrow. Thanks for the discussion!