Although D&D 5e does not provide a medical definition of a disease, from the examples presented it appears that the term is meant to encompass contagious illnesses. For instance, this site lists sight rot, cackle fever, and sewer plague. All of these can be contracted by proximity or contact, suggesting that they are caused by some contagious element, be it viral, cellular, prionic, magical, or whatever.
The terminology used in the description supports this:
A plague ravages the kingdom, setting the adventurers on a quest to
find a cure. An adventurer emerges from an ancient tomb, unopened for
centuries, and soon finds herself suffering from a wasting illness. A
warlock offends some dark power and contracts a strange affliction
that spreads whenever he casts spells.
A simple outbreak might amount to little more than a small drain on
party resources, curable by a casting of lesser restoration. A more
complicated outbreak can form the basis of one or more adventures as
characters search for a cure, stop the spread of the disease, and deal
with the consequences.
"Plague", "contract" and so forth. I would say that these are supposed to be infections. There's no mention of any medical condition that arises from genetics, diet, age, trauma, or other such stimuli.
Obviously, the DM is free to classify any other condition as a disease if they so desire. However, both some philosophical introspection and some consultation with players may be in order. The definition of a disease is controversial: Is disease a deviation from what the patient considers ideal, what society considers optimal, or from some abstract biological notion of correct functioning? Handled poorly, this is a matter that could even end up being hurtful to players. One should probably ensure that, whatever one's personal outlook, both DM and players agree on the definition and mechanics that will be used in-game.