Thanks for your question. While I have answered or tried answering these and similar questions for awhile now, apparently there is something I have not been clear about so it is a good thing that you bring your questions again with an added request for a particular definition. First of all, to act in another's name means to be entrusted by them with some task or responsibility which you do with their authority and while representing them and their wishes and wisdom, their values, actions, spirit, etc. Ambassadors act in the name of a country or monarch, for instance, and they may set many of their own perspectives, values, etc aside in doing so. By the way, while a Catholic lives a Catholic life, and so, lives that life in the name of the Church, this does not mean that everything they do is specifically done in the name of the Church. For instance, if you are an athlete at the Olympics, this does not mean you compete "in the name of the Church". (The flag you would compete under is not the flag that flies over the Vatican!) Still, as a Catholic, to some extent, you would represent the Church's life with your own life, and so, with the integrity of your competition. C 603 hermits are specifically entrusted with the responsibility to live solitary eremitical life under c 603 "in the name of the Church".
This means the Church has discerned the vocation with them and determined that they can serve God and the Church's consecrated life (etc!) publicly in this specific state of life. Such persons have been Catholics, and likely, they have been Catholics AND hermits for some time. But until the Church has extended (and the candidate has accepted) the canonical rights and obligations extended to them by the Church associated with their profession and consecration by God, one has not yet become a Catholic Hermit in the consecrated state, one who lives this life in law and with the authority of and responsibility to the Church (or, that is, to God through the Church's mediation). At the same time, not everything I do is specifically done in the name of the Church (I don't write this blog in the Church's name, for instance), but everything I do or fail to do (and here I include this blog and other writing I do) is part of the way I represent this ecclesial vocation, and so part of doing it well, or doing it badly.
Catholic Lay person AND Hermit:
Baptism alone absolutely means you can (and should) use the name Catholic. You are a Catholic and live as a lay person (a member of the People of God, λαος του θεου), and you are called to do so in the name of God and of the Church. You have been called and commissioned to witness to the Catholic faith and the truth of the Gospel of Jesus Christ entrusted to this Church with your entire life, and you have accepted this call and commission publicly in baptism. This also means you are bound by the same canons (universal or catholic norms) that bind any other lay person by virtue of their baptism. You can also live as a hermit, though privately, without the Church extending the further canonical rights and obligations marking such a vocation; you remain a Catholic AND a hermit (or a Catholic lay hermit, or a Catholic "non-canonical" hermit) unless and until you change that with a "second consecration" under c 603, for instance. Your "standing in law" is as a Catholic lay person, meaning you live your life in the lay state, and your standing as a hermit would then be part of this specific state of life. Hermit life itself, in that case, is not specifically undertaken in the name of the Church, nor would your commitment to hermit life have been specifically received by the Church. It would remain significant, and its significance would differ in some ways from that of a canonical hermit.Non-canonical does not mean illegal!
A bit more about common Catholic usage (hoping this is helpful): non-canonical, despite the sound of the term, does not apply to a Catholic's whole life as though the person is non-canonical or without law, much less illegal, because the laity are all bound by certain canons. Rather, it means a non-canonical hermit lives eremitic life itself without the additional law that becomes binding in some commitments the Church receives and commissions. Also, I should note that the use of lay has two senses, the first is hierarchical. In the Roman Catholic Church, which is hierarchically "ordered", one who is not clergy is lay. Anyone not a priest (or in orders) is laity. This is the hierarchical sense. The vocational sense is a bit different, and it is what I mainly concern myself with in this blog. Here we add the consecrated state, which draws from either/both clergy and laity. There are clerical, consecrated, and lay states of life, and it is possible for one to be a priest in the consecrated state or a lay person in the consecrated state. Men Religious who are not priests/deacons and women Religious are also laity, though we say they are in the consecrated state of life because of the additional ecclesial commitments and canons that define this state of life for them.
Publicly can also be used in two senses:
Publicly is also used in two senses usually, but in terms of vocations, we are not talking about the common sense of "out among ordinary society". "Publicly" here is about being in the midst of and seen by the public. It can mean notorious. But in the Church, to say a vocation is a public one means that it is marked with and defined in terms of public rights and obligations. It is the antithesis of a private state that is not bound by public rights and obligations. Your own baptized state is a public one; it is not a private state marked by a private commitment. So are marriage, religious profession, ordination, etc., though they all have appropriate degrees of privacy. An essentially hidden vocation like that of the c 603 hermit is also a public vocation; that term has nothing to do with notoriety or the amount of time spent out of the hermitage with others. Yes, the rest of the Church (and the larger world) has the right to expect the person to live her commitments. This is because these commitments (and thus, her vocation) are not private even though her vocation is essentially hidden!As you can tell from the discussion of "standing in law" above (the real meaning of the term status) and the reference to the glorification of God just made, the term "status" is not about prestige nor does this kind of standing necessarily indicate a lack of humility. I agree that sometimes people gain prestige from certain vocations and states of life. They are honored for these. In the very best instances, prestige is awarded because the vocations are lived well, serve, and are inspirational for the Church as a whole. Most of the religious men and women I know hate being put on some kind of pedestal, and especially treated like they have some kind of direct line to God. And, though I know some genuinely holy people, they definitely understand that everyone in the Church are called to be saints; the call to holiness is the ordinary and universal call to all persons!
I sincerely hope this is helpful!

