IUPAC nomenclature of cycloalkadienes with substituents and alkenes on the same carbons
The question is based on a false premise:

Generally we try to give just enough information to be able to reconstruct the original compound, but not too much.

This isn’t true. The prime aim of IUPAC nomenclature is clarity and the construction of unambiguous names.

Three logicians walk into a bar
The bartender asks: “Do you all want a drink?”
The first logician says: “I don’t know.”
The second logician says: “I don’t know.”
The third logician says: “Yes.”

IUPAC nomenclature doesn’t work like that. Generally, the complete information about a structure is explicitly given by the name and does not rely on any implied information or hidden logic involving other structures.
The fact that “1-methylcyclohexa-2,4-diene” would actually be called 5-methylcyclohexa-1,3-diene does not change the preferred name for 1-methylcyclohexa-1,4-diene.
Nevertheless, there are a few exceptional cases where locants are omitted, but only when there is no ambiguity. For preferred IUPAC names, the nomenclature rules explicitly stipulate when locants are omitted. In most cases, if any locants are essential for defining the structure, then all locants must be cited for the structure.
Also note that your third example is clearly wrong even when following your unorthodox logic. Your name “1,4-dimethylcyclohexadiene” could mean 1,4-dimethylcyclohexa-1,4-diene as well as 1,4-dimethylcyclohexa-1,3-diene or even 1,4-dimethylcyclohexa-1,2-diene, which are all correct IUPAC names.