There is no hard rule
As you say, the main reason to care about how disease is defined is because game features like Lay on Hands can cure them, and players and DMs want to know what can and cannot be cured.
I think in practical terms, these features can cure
- anything that the game explicitly says is a disease
- anything the feature explicitly says can be cured
- anything the DM determines to be a disease
Unfortunately, there is no explicit or rigorous definition of disease in the game, so the corner cases will be a DM call.
In the case of mummy rot, it says it is a curse, it does not say it is a disease. It also says it can be cured by remove curse, a third level spell. Other magic that can break curses could also cure it.
Lesser Restoration could only cure it, if the DM decided to also treat it as a disease. Doing so would make mummy rot easier to overcome, as Lesser Restoration is only a second level spell (and because other features that can heal disease then also work).
Game Definiton of disease
As cited in this answer to "What Counts as a Disease?", the closest thing we get to a definition of disease in game is given on p. 256 DMG, under the heading "Diseases":
A disease that does more than infect a few party members is primarily a plot device. The rules help describe the effects of the disease and how it can be cured, but the specifics of how a disease works aren't bound by a common set of rules. Diseases can affect any creature, and a given illness might or might not pass from one race or kind of creature to another.
While the term "infect" showcases that infectious diseases are certainly diseases, the rules make it clear that there are no common rules for diseases. Diseases may be contagious, or they may not be. So the concise, sharp defintion you are looking for does not exist.
The section provides further examples:
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 plague is caused by infections, and that seems to be the primariy kind of disease the game is concerned about, as the entire next paragraph is about outbreaks. But a wasting illness can have other causes, such as hereditary disorders that cause muscle atrophy, or chronic wasting disease, that appears to be caused by prions (malformed proteins). And the strange affliction that manifests when casting spells has nothing to do with natural, viral or bacterial infections either. So diseases can encompass more than just infections.
By default we turn to real-world usage if the game fails to define a term. Unfortunately, that does not help so much here either, as even in the real world, what counts as a disease is not sharply defined and is complex.
For example, are infestations like tape worms diseases? Not by themselves, but they can lead to disease. How about extenral parasites, like lice? They generally are not, but there are epidermal parasitic skin diseases, so sometimes they are.
Dictionary defintions also do not help much. Here is the one from Oxford Dictionary
A disorder of structure or function in a human, animal, or plant, especially one that has a known cause and a distinctive group of symptoms, signs, or anatomical changes.
And here from Encylopedia Britannica:
disease, any harmful deviation from the normal structural or functional state of an organism, generally associated with certain signs and symptoms and differing in nature from physical injury. A diseased organism commonly exhibits signs or symptoms indicative of its abnormal state. Thus, the normal condition of an organism must be understood in order to recognize the hallmarks of disease. Nevertheless, a sharp demarcation between disease and health is not always apparent.
Unfortunately, these definitions would include all kinds of harmful magical alterations that can exist in the game world and have nothing to do with disease there (the people writing them did not have to think about exluding magic as a possible source of harmful alterations).
History of Mummy Rot
If Mummmy Rot were treated as a disease, it would be much easier to remove, with a simple level 2 Lesser Restoration instead of the Level 3 Remove Curse.
In general, for power balance reasons, if the game provides a specific spell that allows to overcome something, the principle is that other spells should only be able to also overcome it, if they are of equal or higher level.
For this question, it might be useful to look beyond 5th edition to understand how we should treat mummy rot.
In D&D 3.5:
Mummy Rot (Su)
Supernatural disease (...)
Unlike normal diseases, mummy rot continues until the victim reaches Constitution 0 (and dies) or is cured as described below.
Mummy rot is a powerful curse, not a natural disease. A character attempting to cast any conjuration (healing) spell on a creature afflicted with mummy rot must succeed on a DC 20 caster level check, or the spell has no effect on the afflicted character.
In AD&D 2e:
Mummies are horrific enemies. A single blow from one's arm inflicts 1-12 points of damage, and worse, its scabrous touch infects the victim with a rotting disease which is fatal in 1-6 months. For each month the rot progresses, the victim permanently loses 2 points of Charisma. The disease can be cured only with a cure disease spell. Cure wounds spells have no effect on a person inflicted with mummy rot and his wounds heal at 10% of the normal rate. A regenerate spell will restore damage but will not otherwise affect the course of the disease.
Cure Disease in 2e was a third-level spell like Remove Curse, not a second level spell.
In Ad&D 1e:
The scabrous touch of a mummy inflicts a rotting disease on ony hit. The disease will be fatal in 1-6 months, and each month it progresses the diseased creature loses 2 points of charisma, permanently. It can be cured only by a magic spell, cure disease.
Again Cure Disease in 1e was a third-level spell like Remove Curse.
From a historical perspective, Mummy Rot was a disease, but an especially tough one. It was much harder than in 5e to overcome diseases, and an encounter with a mummy was a frightening proposition for a low level party. I recall one adventure, where one of the PCs was hit by it, and that turned into a whole side quest racing against time to obtain a cure before the character died.