Thanks for your questions. I have written extensively about all of these in the past so I encourage you to look at pertinent posts, especially those in the past month or two. One post you might want to check out is Why God allows Many Forms of Hermit Life. The following article might also be helpful to you: Crash Course in Language. Let me summarize the situation as it applies to you.
You are a non-canonical hermit only in the sense that c 603 and the canons applying to religious (or consecrated life) do not apply to you. However, you are a lay hermit and all of the canons applying to laity apply to you until you are admitted to profession (a public act changing one's state of life) and then consecration as a c 603 hermit or member of an eremitical institute of consecrated life. Once that happens canons are added to those already applying to your lay life. Both states, lay and consecrated, are governed by canon law. To call one a non-canonical hermit is a shorthand way of saying the additional canons do NOT apply to them (yet). It does not mean that hermit is illegal since you are free to live as a lay hermit while honoring the canons that apply to all laity.
If you decide God has called you to live as a consecrated hermit, you must apply to be admitted to a process of MUTUAL discernment and formation with your diocese. Because this is an ecclesial vocation and belongs to the Church before it belongs to you, the Church herself must discern that you are indeed called to this. Our own discernment is important, but it is not sufficient when dealing with ecclesial vocations. This is a difficult point for most folks to "get". Meanwhile, if, in time, the Church agrees with your discernment and also that you are ready for public profession, then she will admit you to temporary profession and eventually to perpetual profession and consecration as a c 603 or diocesan hermit. At that point you accept the rights and obligations of a public and ecclesial vocation and MAY (i.e., are permitted by the Church whose vocation this is first of all to) call yourself a Catholic Hermit. This is because you would then have been called by God and the Church to live this vocation in the name of the Church.Currently you are a Catholic AND a hermit. Another way of saying this is to say that you are a (Catholic) lay hermit (that is, you live hermit life in the lay or baptized state alone as you are entirely free to do), or you can say you are a non-canonical hermit because you haven't (yet) been admitted to eremitical life in the consecrated state; for this reason, the additional canons of consecrated eremitical life don't apply to you. You are entirely free to call yourself Catholic; you are entirely free to call yourself a hermit (if you are one); you are even free to say you are aspiring to becoming a Catholic hermit if that is also so, but you are NOT an illegal hermit or somehow not Catholic. To use the term illegal is demeaning of yourself and all Lay Hermits. Please let go of it!! You can read more on this or get back to me if it raises additional questions or confusions. I sincerely hope it is helpful.
N.B., a reminder that the term lay can be used in two different senses. In the hierarchical sense anyone who is not ordained is lay or part of the laity (from the Greek laos (λαος) for People as in People of God or laos tou theou). This sense includes religious and consecrated persons. In the vocational sense of the term lay, it is one of three basic states of life: lay, consecrated, and clerical. We have hermits in all three states of life and honor them all. In using lay above I am mainly speaking about the vocational lay state. For the present, and until a diocese admits you (or anyone else living as a hermit) to profession and consecration, you (they) are a lay hermit in both hierarchical and vocational senses of the term lay/laity.