Using a microwave oven transformer as ballast by shorting the secondary
A microwave oven transformer (MOT) is not (despite the name) an ordinary transformer, it is a specifically designed non-ideal transformer. 

Ideal transformers are made as 'perfect' as economically possible, so highest winding inductance, highest permeability core, and lowest leakage inductance possible. Non-ideal transformers have one or more of those things finite, in a designed way, for a purpose.

A MOT is designed to have a relatively large leakage inductance, in order to resonate with and reduce the impedance of the doubler capacitor it drives. This is achieved by using magnetic shunts between primary and secondary windings. You'll see these referred to in the articles you've linked to. Another way to describe finite leakage inductance is that the coupling between the windings is loose rather than tight.

When you short circuit the secondary of a MOT, the primary presents essentially the leakage inductance. This will limit the current drawn to 2 or 3 times the normal operating current, the right ballpark for 'fun with MOTs', and generally low enough to leave fuses/breakers intact. If you leave the secondary open, it presents the primary inductance, too high to be of any use as a ballast. This is what you'd get if you removed the secondary from the MOT.

If you take an ideal transformer and short one winding, then the other winding will also be effectively 'shorted', as the leakage inductance is very low. This will not work well as a ballast.

Be aware that a MOT is the worst possible source, from a safety point of view, for starting your electrical or high voltage education. If you get bitten by an auto ignition coil, you will live. If you get bitten by a neon sign transformer, you may well live. If you get bitten by a MOT, you will probably die. It has enough voltage to jump through dry clothes, and enough current 10 times over to stop your heart. Do not be casual around MOTs.

It's worth pointing out that your diagram is at odds with your description. The diagram shows a ballast in series with the secondary, not the primary. The correct place to use a 'shorted secondary MOT' ballast is in series with the primary, which is where the resistive ballast is correctly shown. Remove the resistor, and replace it with the primary of an 'shorted secondary MOT' ballast.

You could, if you wanted to live more dangerously, use a MOT ballast in the secondary as shown in (a) by using its secondary, with the primary short-circuited. While this is electrically more or less equivalent as far as the arc is concerned, it means the case of that ballast is live, and that any other discharges drawn from the MOT secondary are not ballasted. A primary-side ballast as in (b) is by far the better mode of operation.