My intuitive response to the above question is to consider that the difference between the meaningfulness of "melancholy" vs. "melancholic" has everything to do with speaker's context and the social distancing and/or intimacy of register, rather than adjective-ness or noun-ness.
My first encounter with the the word "melancholy" was in 1972 (I was four) as I read it in an arrangement of a score for voice and piano of the folk song "Down in the Valley". The term "melancholy" was a notation to indicate the dynamic or mood with which the song should be sung. (This would match the questioner's citation in the first definition: "sombre".) From that time I had always understood this term to mean a particular mood - one that is connected to an ancient/universal mourning that is both a darkness of grief and an affirmation of loss...some call it feeling "blue" or even feeling "a bit of the blues" - again the context of song lends to the meangingfulness of the term. I believe that when expressed as a state of condition by the speaker, "I am feeling melancholy," or "Sometimes I have bouts of melancholy," it is an intimate expression of experiencing something deeply within and greater than oneself at the same time; "melancholy" has a sense of intimacy and mystery to it. My husband and I have kindred-meanings for the term: when he says he's "a bit melancholy, but for no reason," I know that nothing specific occurred that caused him to feel that way (nor that I had done something wrong to make him feel that way) and likewise when he notices I'm feeling down and I want to assuage his concern that something is really wrong, I just tell him I'm a bit "melancholy" today.
However, even though I attached personal meaning to the term "melancholy" as a child, I learned as I got older that there is a psychiatric/pathophysiological application of this term that has a very different kind of significance when it is used to ascribe a behavior, action, or state of mind/being to a diseased or disordered individual. For me, this is where the the use of "melancholic" breaks completely from "melancholy"; the use of "melancholic" as a clinical term is not unaffiliated with the impact of social stigma on those suffering from mental illness. (Again, as in the questioner's above cited second definition: "sadness like an illness".) In the article "Issues for DSM-5: Whither Melancholia?" appears the phrase "melancholic patients respond better to..." and later the author states "melancholia is a lifetime diagnosis, typically with recurrent episodes..." (See the article online at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3733615/). Clinical depression runs in my family, and my parents, both professionals in the medical field, never used the word "melancholy" or "melancholic" at home, as I think they felt those terms implied a diagnosis, even in social settings. I think the use of "melancholia" as a term to describe a psychological disorder (a noun) and the purposeful variation of that term as (an adjective) "melancholic" to ascribe a patient's condition (and recommendation for treatment) creates a medical/physical/social distancing in register that also creates a distinct difference in meaningfulness.
(I did notice in one of the responses above, in that great historical citation the use of "melancholy" to describe the patient's circumstances, and not the patient himself...perhaps that is the reason to not use "melancholic": to ascribe the behavior to external circumstances and not the individual, thereby creating pretext for "vindication" and "extenuate the crime in him"... I do not know enough about the historical/legal context of mental health treatment in the early 1700's to guess at difference in usage back then, but for that speaker's purposeful use of "melancholy" - being an external condition beyond the power of human control - as stated in the argument that "God deprived him of his senses" seems a form more specific to the speaker's goal. In this regard, I think there is a pragmatic difference in usage: "melancholic" which could describe a condition that is internal, conditionally symptomatic, melancholy-like, and within an individual's power to control; and therefore not enough of an excuse for a "strange and deplorable frenzy".)
In his book, "Adios, Strunk and White", Gary Hoffman describes how his studies in architecture and the Bauhaus design theory "form follows function" influenced his approach to literature. For me, this notion of form and function applies to how I would make the choice between "melancholy" or "melancholic":
"Melancholy" = evocative (describing mood)
"Melancholic" = prescriptive (ascribing condition)
1) "Looking through old photo albums always makes me feel a bit melancholy."
2) "I had bad dreams and tossed and turned all night - my lack of sleep has put me in a melancholic mood this morning."