WARNING: DON'T USE LITHIUM BATTERIES, or any type of laptop battery or phone battery. When connected to a small coil of wire, they may catch fire or explode. Even an alkaline battery will heat up dangerously after one or two minutes.
Yes, you can briefly short out a small alkaline battery or carbon-zinc battery without destroying it or causing a fire. But the battery runs down very fast, and heats up quickly. Hint: use a push-button rather than a toggle switch. That way the battery is turned off when you're not holding down the button.
In his video, he really should be using an AA-cell or perhaps a C-cell. He'd still get about the same current, and perhaps a bit longer lifetime for each battery. It's much cheaper to run down a bunch of C-cells, instead of using expensive 9V batteries.
The coil in his video is a short circuit. A few meters of thin wire will have a resistance of less than an ohm. One ohm of #30 gauge wire gives three meters, one ohm of #26 gauge wire gives 7.5 meters (His wire appears to be thicker than #30 gauge.)
The Amazon.com link, those batteries would work fine. Also try AA cell or C cell, plus a plastic battery-holder for those size.
Also see: small coils projects, motor and generator
This one: high speed generator, powered by bow-drill