If the amperage rating of a wire is not exceeded, such as is the case in the house/heater example, the wire may get warm--not hot.
In Bleach's lawnmower's case, he said the battery wires got HOT. That will not occur unless they're overloaded, such as a dead short.
First of all, I can see where you are coming from.
Second, one man's warm is another man's hot.
Third the main wires are rated at 100A continious .
The only thing that can supply more than 100 A is the battery
So by following your over current load logic, it has to be a short in the main power wires because any other wire passing enough amps to heat the main power wires would melt or blow the fuse.
But the fuse does not blow and problem only happens when the mower is running.
There is no part of the main power supply that is in circuit engine running and out of circuit engine off.
The only time you get a heavy load is when the solenoid is energised and if the solenoid was shorting to ground after being de-energised, ( engine running / starter off ) it would stay short, ignition or or ignition off.
Electrically the only difference mower running to mower not running is the secondary power supply coming from the alternator.
Thus if the problem only happens when the engine is running the most logical source of the problem will be the charging system.
Now batteries get hot when one of 2 things happen
1) connected to too high a voltage ( 16V +)
2) connected to a reverse flow of current (Wired backwards or getting AC )
If the battery plates are getting hot, the battery terminals will get hot .
If the battery terminals are hot then then heat will also travel back through the terminals and into the wires so they will get hot.
As mentioned earlier this was very common in the 30's to the 60's with Lucas dynamos fitted to motorcycles that would reverse polarize when left not in use for a long time, flatten the battery and then boil it or explode it when the bike was running by pumping current backwards into the battery. And we are talking 5A to 10A on a 6V system or 3A to 6A on a 12 V system.