How did I screw up this rice pudding recipe?
I've never made a baked rice pudding before but I have a couple of guesses at what went wrong...
1. Chosen rice preparation method caused loss of needed starch.
First, I don't think the type of rice matters much but I will say that your method differs from the standard method in the US for cooking rice. Generally, we boil the rice with about twice the volume of water as rice until all of the water is absorbed. There's no dumping out extra water or rinsing.
This may not make the best rice for direct consumption but it concentrates the starches because none of them escape when you drain off the water or, worse, rinse it.
This starch is probably at least some of what helps the rice thicken while being baked... so your soupy result is because there wasn't enough thickening agent.
2. Stiffly beaten egg whites means something special.
I can't stress enough, when cooking, if you want to be successful, look up any terms you don't understand... or come and ask us here... that's what we're here for.
Regular egg whites look like this (all images taken from the Joe Pastry link below):

Egg whites that have been &quot;stiffly beaten&quot; look like whipped cream. They have peaks that stand up on their own when the whisk is removed from the bowl. This Joe Pastry page has a guide to the method for achieving this.
The ones below are soft peaks - see how they fold over at the tips?

And here is your target, &quot;stiff peaks&quot;. No folding tips, dense foam.

If your egg whites looked more like the first picture, they won't do their job of adding structure to your pudding.

Stiffly beat egg whites and fold into mixture.

The word &quot;fold&quot; is important here. To avoid breaking the egg whites back down into mush, you gently fold them into the pudding so that they retain their structure for their time in the oven.
So, I think that these two issues will be your main culprits for why this recipe failed on you.

Additional possible but less troublesome issues:

Milk percentage - The milk percentage probably didn't matter but it's probably going to be better in the end if you use whole milk. My go-to recipe uses whole milk and half and half (and no eggs), so it's much richer than this one... it's also a stovetop method and it's amazing.

Egg size - The standard for eggs in recipes in the US is usually &quot;large&quot; sized eggs (~2 oz/~56 g) unless otherwise noted, so if you used medium in the US or Canada (~1.75 oz/~49 g), that will make a bit of difference as the egg is your main thickening agent in addition to the starch - but if you're in the UK or Europe, I believe the equivalent for that is &quot;medium&quot;, so that shouldn't be an issue. Info on egg sizes in different countries on Wikipedia.

Oven temperature - Your oven could be off but if you find it generally performs well, this likely isn't the issue, particularly considering all of the other issues.