Right. If someone other than you wants to see, then the external device is the best idea. I'd also get a model that has a battery so the information isn't lost if it's unplugged from power.
If the Leaf charger was hot around the plug, there had to be some kind of problem with the charger's plug. Probably a bad connection where the plug blades connected inside the plug to the charger's cable. That's a manufacturer defect. Being a 2013, I'm guessing you're not going to get Nissan to warranty replace it, but it doesn't hurt to tell them about the situation and hope they swap it out.
That whole "Don't use an extension cord" thing is BS. Just use the right extension cord and you'll be fine. It's only if you use a cord that can't handle the current that you run into trouble. The whole house electrical system is a giant bunch of electrical extension cords in the walls. Since the cord you used was new and 15a rated, I doubt that was the bottleneck.
One thing helpful to make sure you know what kind of power you're actually dealing with is the Kill-a-watt. https://www.amazon.com//dp/B000RGF29Q That will let you see exactly how much power you're pulling. It shows Watts, Volts, Amps, and even tracks how many KWh you're using over time. With that you can see if the charger is pulling 13A which is fine, or 15.1A which is very much not fine.
I plugged my leaf into a power strip with a breaker switch in it to put in a layer of protection if it sucked power too hard.
Kill-a-watt (120v)