Does starting the engine with one of the spark plug wire disconnected damage the starter coil?

No. The reason your other friend incorrectly said that is because if a charged plug was touching a ground (the frame or whatever) that's a short over a device designed to produce a high voltage, which, obviously, could ruin it. However, the plugs cable ends are "highly" female in that there is plenty of rubber to prevent them from accidentally touching a ground. You can't even touch the metal inside the cable with a finger.

From the coil's perspective, the electrical difference between sparking in a spark plug and sparking through the plug cap rubber and onto the engine isn't that much, except it has to go over a much larger air gap distance. The main point is that it isn't being shorted because of the design. Cars (where most of the engineering came from for your motorcycle) are > 100 years old. Engineers thought of this exact situation already and designed accordingly.

That said, you might have died had the loose cable had touched you while you were driving 50 mph. Unlikely you'd be doing that missing a cylinder of power, but that's a bigger concern in my book.

Could this have done damage to the bike?

One thing that can happen on all engines having multiple cylinders and a catalytic converter is this:

1. N-1 cylinders are firing, providing hot exhaust to the catalytic converter
2. The remaining cylinder is not firing, so the raw evaporated fuel goes to the catalytic converter
3. The fuel burns in the catalytic converter that's at the reaction temperature
4. Your catalytic converter overheats and is damaged

I used to have a faulty connection to one of the spark plugs on a 1989 Opel Vectra. So I know that temporarily running the engine on three cylinders firing won't ruin either the ignition coil or the catalytic converter -- assuming you immediately notice the condition and fix it.

I also once suffered an ignition coil wire disconnection, which obviously stopped the engine. Fortunately was easy thing to fix, in two minutes I was already driving again. This momentary event damaged neither the ignition coil nor the catalytic converter.

However, starting to genuinely drive at a fast speed on a car having one cylinder misfiring could damage the catalytic converter. I understand that new cars automatically detect this condition and stops fuel injection to the misfiring cylinder.