Yep, coiled can get warm, stretched out, not so much.
I did it once myself without thinking. Just a dumb moment.
Plugged an electric heater into a mostly coiled cable reel and 15 or 20 minutes later realised that I'd done something stupid.
When I unspooled the cable the bulk of the flex from inside the coil was too hot to touch and super floppy.
To do some numbers:
In the UK you can buy a 50m extension cable, good commercial brands, no problem.
That is 100m live and neutral return.
In 2.5mm sq extra heavy flex the wire has a resistance of 0.74 ohms
Carrying its rated load of 13A from a standard plug, the cable drops 9.6V along the wire and dissipates 125W (V=IR, Power = IV = I squared R = V squared/R)
Remember the heat put out by an old style 125W lightbulb. Not so much spread out along 50 metres but imagine all that heat trapped inside the coil of wire. It will get very hot very quickly.
Do the same for a lightweight cable reel, same length but 1.5mm square and 10 amp rated (though it will be on a 13A fused plug so relying entirely on the user not to overload it)
Now we have 1.21 Ohms resistance, 12.1V dropped along the cable and almost the same heating, 121 W even though the current is less. Again the cable will quickly get hot if coiled up.
Electric heaters are the killer.
Most of our tools only run for a short time. Anything that runs continually at high power for a long time poses the greatest risk of heating and fire.
We should teach ourselves to be nervous of electric heaters, immersion heaters, and to a lesser degree powerful pump and fan motors because these are big loads that can be left running for hours at a time.
Oh, and this is why the way a cable is installed must be taken into account when you design household wiring.
These dissipation numbers aren't much different for fixed wires, so you could easily have 125W of heat being generated in a long run of twin and earth. No big deal if that wire is in free air to cool it, but buried under 6 inches of loft insulation a wire can become noticeably warm even if it is stretched out and not concentrated on a reel.