[[Dear Sister, if I choose to make private vows as a hermit, do I need to check with my bishop? Do I need his permission? Would bishops know the lay hermits (I mean the privately vowed hermits) in their diocese --- maybe because my pastor tells the bishop about me? I'm asking because of the following comments by (a lay hermit): "The choice may include the individual with spiritual director or if would-be hermit in a religious order with a superior. And as a Catholic in one manner or another, at some point in time, one's bishop is involved more directly or else indirectly, whichever the path of eremitic is discerned and taken." She also says she usually trusts her pastor to tell her bishop about her but does not need to get permission to move into a parish and live as a Catholic hermit because she has private vows.]]
Similarly, unless one already knows one's bishop in some capacity chances are slim to non-existent that he will know of one's private vows. The Church does not keep any records of such private commitments precisely because they are private, not public and not undertaken in the name of the Church. Generally speaking, bishops may come to know of lay hermits (hermits in the lay or baptismal state alone) living in their dioceses but chances of this are greater if the hermits are sources of confusion or difficulty --- or, alternately, if these persons are particularly edifying or inspirational to their parishes and pastors. Another instance in which privately vowed (not professed since by definition this term implies a change in state of life) hermits may become known to their bishop is if such a person decides to seek public profession under canon 603. (In this instance the bishop may truly become "involved" in a real way and not simply know of the person.) Living as a lay hermit either with or without private vows can be an important way of preparing for a publicly professed life; in fact, unless one is already a religious, living as a lay hermit for some meaningful period of time (at least several years) during discernment and formation in anticipation of public profession is necessary. No one is professed under canon 603 unless they are already hermits in some essential sense.
I have waited to respond to the last piece of your question because it really raises significant issues of irresponsibility in claiming the name "Catholic hermit." Remember that a diocesan bishop is responsible in a direct way for anyone or anything in the diocese which is explicitly identified as "Catholic." A Catholic institute of religious, a Catholic priest, a Catholic broadcasting organization, a Catholic theologian, etc, etc. All of these claim to act in the name of the Church and as such, they do so under the specific authority of those commissioned by the Church to exercise such authority. In diocesan organizations or entities this is ordinarily the diocesan bishop. Again, no one may adopt the designation "Catholic" unless they are specifically authorized to do this. Canon Law is clear about this and the Church acts on significant instances of abuse where the name Catholic is being misused and/or misrepresented. (Ordinarily a single lay hermit mistakenly calling themselves a "Catholic Hermit" will not fall to the bishop to handle; it is more likely to be addressed by the parish priest or pastor.)
Baptism gives us the right to call ourselves Catholic but it does not give us the right, nor convey the obligations linked with being a Catholic Hermit --- a Catholic who lives eremitical life in the name of the Church and thus, under the specific supervision of the diocesan bishop. Remember c 603.2 reads, [[§2. A hermit is recognized by law as one dedicated to God in consecrated life if he or she publicly professes in the hands of the diocesan bishop the three evangelical counsels, confirmed by vow or other sacred bond, and observes a proper program of living under his direction.]]
It is irresponsible of one to claim the title "Catholic Hermit" and do so without being specifically and publicly (canonically) commissioned to do so by the Church. However, it is equally irresponsible for an individual claiming to be a member of the consecrated state of life and calling themselves a Catholic hermit to fail to contact the bishop of a new diocese directly. They do not hope or expect or count on someone doing this for them! They do not trust it to chance. Members of Catholic Institutes of Consecrated life know this and work out moves in location with those in legitimate authority. Consecrated virgins also know this, not least because it is specifically required in the guidelines codified for them by the Vatican. Solitary Catholic (c 603) Hermits know it because c 603 is clear re who is actively supervising their vocations!
Consecrated life is constituted by several different ecclesial vocations (canonical cenobitical, canonical eremitical, consecrated virginity, etc) and if one really is called to such a vocation one honors the Church and authority in the Church which has publicly professed, consecrated, and commissioned one. After all, ecclesial vocations belong in a special way to the treasure and patrimony of the Church. Thus, C 603 hermits must get approval of the bishop in the diocese to which they wish to move if they wish to remain a professed hermit. Otherwise, should they move, their vows cease to be valid due to a material change in the substance of those vows.
At this point the hermit would cease to be a Catholic Hermit and, if she continues in the eremitical life, becomes a Catholic who also happens to be a hermit with an entirely private commitment. Such a life is significant but it does not constitute an ecclesial vocation, is not lived in the name of the Church, and thus, is not specifically called by the term Catholic. The bottom line in all of this remains: solitary Catholic Hermits live under the direct supervision of a legitimate superior, namely the diocesan bishop; if she moves she will have contacted the new bishop directly re continuing to be a solitary Catholic Hermit before she makes her move, and again afterwards if he agrees to receive her as a diocesan hermit who will be living eremitical life in the name of the Church. Private vows are a possibility for her if the bishop will not receive her as a canonical hermit under c 603, but she cannot continue to represent herself as a Catholic Hermit.
I sincerely hope your own discernment in eremitical life goes well. If you should decide to make private vows as a lay hermit (again, a hermit in the lay state) I wish you well in that especially. It is not easy living such a life without the moral support of one's parish family (though you can certainly secure that if you desire it) and the Church at large --- at least that is what I have found. As a lay hermit you may well have the support of members of your parish and find that you have something unique to offer the parish at large as well. As a lay hermit you will have something to offer the post Vatican II Church as well because this Church is still trying to implement Vatican II and its esteem for the laity. Because you will represent the desert Fathers and Mothers (who were lay hermits and a powerful prophetic presence for and in the Church) you may serve members of your local Church in ways canonical hermits might not be able to do. I hope so!