TIP-Barrel test of oscilloscope
Yes, that metal section of your probe is ground, and yes, minimizing loop area when probing fast signals is good.

If you're probe didn't come with one of those nice little ground springs as shown in the second picture, yes, you can build a makeshift one with a small piece of wire. Keep the wire as short as possible, and keep the distance between the point you are probing and ground as close as possible i.e. ground the probe to the closest ground on the board.

Edit:

I went hunting for ground springs, since I realized that I actually had no idea if you could buy them. The most common term is "ground spring", or sometimes "short ground clip", and individual replacements can typically be found by searching " ground spring" or similar. For instance, Digikey has the following (among others):


BK Precision 5mm ground clip, set of 4 3.6USD, but they're not in stock.
Teledyne Lecroy ground pin 8USD for one, but they're in stock.