Let's put it this way. Genre is a term used to refer to categories of
art/communication that are defined by specific shared characteristics.
Subcategories within a genre are referred to as subgenres. The term media/medium
refers to specific modes of art/communication in which the
shared/contrasting characteristics aren't simply categorical difference but
technological differences in how the art/communication is experienced by the
participant, just as in the physical/technological experience of reading a book
is different from the physical/technological experience of viewing a
film.
So as an example, let's look at this in terms of literature. The medium for the
transmission of literature is the book. That's the physical/technological
artifact through which literature is conveyed. Not all books are literature
however. So do we refer to literature as a genre of books? No, that doesn't
really work. Literature is an artform. Artform is a category of communication in
which the act is expressive and the meaning is implied rather than stated
upfront. Within art and literature, a genre is characterized by a set of shared
artistic techniques, as poetry is characterized by the use of artificial line
breaks, rhythm, etc. Genres of literature include poetry, drama, fiction, etc.
British lit, American lit, etc. aren't genres either; rather they're categories
of literature as art forum not based on inherent characteristics within the art
but on national origin.
So here's how I would categorize anime. Animation is an artform combining the
medium of film and fine art. If we classify animation according to
national origin, then anime is a Japanese artform, and cartoons are an American
artform, as subcategories of the classification artform. If
however, we are referring to specific artistic characteristics shared by anime
or cartoons, then those become genres of the animation artform, with
further subgenres including shoujo, mecha, etc.
So whether or not anime is a genre comprised of subgenres depends on how one is
referring to anime at a given time. If one is arguing that anime has
specific artistic characteristics that distinguish it from the specific artistic
characteristics of cartoons, then anime and cartoons become genres
instead of just national artforms.
I think one of the reasons anime fans have had trouble with this issue is
that we've lacked the word artform in the debate. We need the middle term in
between medium and genre, because the term medium does not really apply and the
term genre has specific parameters that don't apply in every discussion.