Will a battery short out when the terminals are directly connected together or will internal resistance save it?

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

Here is a simple way to look at it. Batteries have internal resistance. When you short circuit a battery externally, the current will be limited by the internal resistance. If the internal resistance is high, the short circuit current will not be too large.

Older, non-rechargeable batteries tend to have high internal resistance. The carbon-zinc battery you linked to will probably have high internal resistance.

Even small lead, NiMH and lithium ion/polymer batteries have very low internal resistance. If you short one out you may get a giant spark and have lots of heat. So don't short any of these types of batteries!

One other thing. The wire of the electromagnet does have some resistance. So the short circuit current will be somewhat limited.

This very much depends on battery type and rating. Also, keep in mind that a coil of wire still has resistance, and is not a real short-circuit. Could very well be some batteries are able to deliver that much current.

Every battery has internal resistance - how much that resistance is depends on a lot of things (type of battery, age of battery, construction, how much charge is still in the battery, temperature, etc). This internal resistance can be more than sufficient to protect smaller batteries like (certain) coincells and certain AA-style cells from damage when shorted.

However, other types of batteries like Lithium-based ones (especially the rechargable types) can have internal resistances in the few milli-ohm range - shorting them out can (and usually will) damage or even destroy the battery.

You can try and calculate the resistance of the coil you will use (or measure it, though that is not straigh forward given the usually quite low impedances involved) and see if your battery is rated to deliver that much current.