Of course, with a definition like this, the big questions become "What does it mean for something or its interests to matter morally for one's own sake?" and "What does it mean for something to be wronged?" In practice, of course, philosophers don't tend to ask this question, which is surprising; or perhaps not surprising, since the most natural answer to the first question is, "They can be considered in good moral reasoning as an end or valuable" and the most natural answer to the second is "Something is wronged when its value is not properly regarded". These answers, though, would make talk of moral status otiose -- it would contribute nothing that could not be said better another way, and almost anything could have 'moral status' in this sense at some point or another. What philosophers in practice use 'moral status' to do is to argue that things should be denied rights, and answers that give everything moral status, even if only contingently, even if only intermittently, would make 'moral status' nearly useless in such arguments. (It would still be possible to have such arguments, but 'moral status' wouldn't be very handy for them.) Yet there is very little reason to deny that things can morally matter for their own sake at least some of the time; and almost any account that did deny it would end up being very difficult to apply consistently without the kinds of actions that many people would consider egregiously unjust.
I think that virtually all discussions of 'moral status' are nonsense of the worst sort. The article, however, is quite good; it lays out the common positions fairly, and while it does not consider the whole problem head-on, it does consider part of the problem. As the authors state (FMS means 'full moral status'):
Nonetheless, providing an adequate theory to account for the FMS of unimpaired infants and cognitively impaired human beings without attributing the same status to most animals has proven very difficult. In fact, our survey in section 4 suggests that this challenge has not been met by any of the existing accounts of the grounds of moral status. Some philosophers have, as a result, questioned or even abandoned this seemingly commonsense view, including the aspect that holds all adult cognitively unimpaired human beings have FMS (see the end of section 4.1).
Of course, we could also accept the point instead of trying to save our accounts of 'moral status' by insisting that not all "adult cognitively unimpaired human beings" have full moral status, and conclude that accounts of 'moral status' don't really contribute anything of importance. But then we couldn't have a cottage industry of people just making up criteria for 'moral status', completely disconnected from common sense or any other way of determining whether they are even heading very roughly in the right direction.