Hi, and thanks for the question. The answer is simple and very straightforward. No, what I write is my opinion, though I certainly strive to be sure that those opinions are well-informed and accurate in theological, pastoral, and ecclesial terms. Neither I nor any other hermit has the authority to write in the name of the Church unless this authority is specifically granted in the giving of what is called a mandatum. However, what I and other canonical hermits have been given is the authority (and more fundamentally, the obligation) to live solitary eremitical life in the name of the Church. To some extent that implies the authority to explore this vocation, to grow and mature in it, for instance, and even to share what we have come to understand with others, but it does not mean that I, for instance, write or speak in the name of the Church. Of course not.
Remember that c 603 is the universal Church's norm for solitary eremitical life. (Other canons apply to hermits living their vocations within a communal context.) It describes a vision of this form of eremitical life and adds to that the requirement that the c 603 hermit writes her own Rule of life based on her experience, education, and training. The diocesan hermits I know take this vision seriously and (to greater and lesser degrees) share with one another so that day by day we may grow in our embodiment of the vision c 603 represents. I tend to share based upon my own experiences, education, training, etc., on this blog and that means that what I write is a reflection of what I have come to know about this vocation and the way I am called to live it. Still, while what I write may be helpful to some folks (including members of the hierarchy or canonists reflecting on this vocation), it does not mean what I write itself is necessarily normative, authoritative, or done in the name of the Church.
The phrase, "in the name of the Church" comes from the idea that the Church commissions people to act in certain ways on her behalf and thus, with the Church's authority. Just as Kings, for instance, charged ambassadors with tasks to carry out in the King's name in various foreign kingdoms --- meaning where the ambassador spoke or acted according to orders, the King himself spoke --- so too does the Church commission each of us to proclaim the Gospel in ways that are appropriate to our own state of life. In fact, each of our lives is meant to be an embodiment of such a commission. Thus, each of us at baptism assumes the name Catholic. We are Catholics who live our vocations in the name of the Church. Those who are married in the Church are also commissioned to live married life in the name of the Church and to do this faithfully -- a rather important charge or mission! Religious do the same with religious life. And so forth.The shorthand way of pointing to all of this is to say of someone, [[X is a Catholic nun (or priest), or is part of a Catholic (or Sacramental) marriage.]] I am a Catholic (diocesan) hermit, which is the shorthand way of saying the Church has commissioned me to live this specific vocation in her name. I do this in faithfulness to c 603, its traditional context, and the larger context of eremitical life. Simply to say one is a Catholic is to claim to have been commissioned to live a foundational vocation to discipleship in the Church's name. That commission was associated with our baptism and is renewed at every Mass as we are nourished in our faith and then sent forth as Christ's own disciples. Beyond baptism, however, we may be called to religious life (including semi-eremitical life), consecrated life as solitary hermits or virgins. These are ecclesial vocations which the Church herself directly entrusts to individuals through a second consecration and commissioning to live in the Church's name. Thus, these persons are Catholic hermits, nuns and brothers, and virgins, not merely Catholic AND a nun, brother, etc.
We see something similar when a theologian is given a mandatum by the Church to teach and do theology in the Church's name. In order to do this, a theologian must do their terminal degree at a pontifical institution and be granted the authority to call themselves a Catholic theologian. This authority can be withdrawn by the Church as well, something that once happened to Hans Kung. Note well, however, simply because a theologian cannot call themselves a Catholic theologian, this does not mean that person's theology is not faithfully and exhaustively Catholic. The same is true of non-canonical hermits. While they cannot call themselves Catholic hermits (i.e., because they are not authorized to identify themselves as living a normative (canonical) vocation), this does not necessarily mean the person is not faithfully Catholic. It simply means the Church has not authorized her to live eremitical life in her name.
Given your questions, what might also be clearer to people reading the posts on this blog over the past couple of months is precisely why I might be upset at someone calling themselves a "hybrid hermit" and not taking as seriously as the Church expects, the commission associated with being admitted to profession as a diocesan hermit under c 603. Acceptance of profession under this canon is associated with an acceptance of a commission to live the terms of the canon as faithfully and fully as one can. The vocation is not an excuse for doing active ministry. The central elements of the canon are normative for this life. If one lives a c 603 life honestly, then they will be hermits, not some form of "hybrid hermit" or "hermit monk". Yes, in time, they will grow into this life more and more deeply if they have truly given themselves over to the vocation the canon describes and defines, but they will never be (called to be) something other than a hermit. If one cannot answer this call or accept this commission to live this vocation in a normative way in which both the hermit and c 603 life itself thrive in them, then one will naturally (and rightly) conclude they are not called to this vocation.