I can see flaws in the method. Adding water by 'depth' is pretty inaccurate unless you use the exact same rice quantity in the exact same pan as the recipe author. Also, bringing it slowly to the boil like porridge is very inaccurate for your timings.
Let's try it a slightly different way. Use ratios & a flash-boil.
There's no absolutely ideal 'works for everybody' ratio as it depends on pan thickness, tightness of lid & stove heat. You'll end up between 1:1.5 and 1:1.75
Though this takes a lot of text to explain, it's actually really, really simple to do & once you have your ratios right for your kitchen, it will 'just work' every time.
Put the kettle on.
(This works best if you live in a 240v country, otherwise you might have to wait a few more minutes before you start 2;)
Using a small-ish saucepan with a tight-fitting lid & the heaviest base you have, pre-heat at full power as the kettle's coming to the boil. The idea of this is that the pan is hotter than the water as it goes in… but not on for so long it's glowing, or you can smell burning teflon;)
Add 1 cup of rice to the pan as it's heating (cup, mug, measuring jug, doesn't matter so long as you use the same measure for the water. That way we're working percentages/ratios not inches). Don't bother to pre-rinse unless your rice is particularly dusty. Supermarket rice really doesn't need it these days, & pre-rinsing will change the flash-boil reaction.
Add salt now if you want salt.
Add a cup and two-thirds (middle of your ratio spectrum) boiling water. The water & rice combo will immediately flash-boil almost out of the pan. Immediately drop the heat to minimum (assuming gas, or take the pan off the ring for a few seconds if electric or anything not instant)
Stir once with a spatula, just enough to know the rice didn't clump, which it shouldn't if the water flash-boils on contact.
Put the lid on. (First time, until you get used to this, keep an eye on it to make sure it's not going to boil over as the initial heat drops away)
Simmer for 15 mins, without lifting the lid to check.
Don't be tempted to lift the lid to see if it's done…
This is a Schrödinger/Heisenberg* method - see below;)
This depends on your pan thickness & how low your ring goes. You learn this through repetition; after that, time it.
While you're learning the timing, then you'll have to just lift it slightly & briefly every minute from 12 or so. Once you've got the ratios for your pan/stove combo you don't do this.
Whilst you're learning at the lid lift stage, you're listening for a 'crackling' sound. That's the indicator your very last bit of water is evaporating off the pan base.
Switch off the gas (or for electric switch off & move the pan to a cold ring)
Leave 15 mins more. Don't lift the lid at all.
We don't need Schrödinger for this bit - the cat is alive;)
'Fluff with a fork' as they always say - or if you got it just right, a quick once-round with a spatula will do it. it won't have stuck.
Profit… or adjust your water ratio/timing/heat for next time.
A lot of this is learning your specific pan/stove. If your ring is hot or your pan is thin, you have to adjust your water ratio up slightly, so you arrive at the 15 minute mark as it crackles. This can mean you end up with a ratio somewhere between 1:1.5 & 1:1.75 If you err, err on the side of dryer.
Glass-lidded pans can take a lot of your early guesswork out of it, as you can see what's going on.
*Schrödinger's cat is a famous thought experiment. You don't know if the cat is alive or dead until you open the box.
We're additionally hampered in this by Heisenberg's Uncertainty Principle - that by the mere act of observing an experiment, you change the result.
Hence, you cannot know if it is done until you lift the lid… but by lifting the lid to see if it's done, you change the time it will take & the amount of water required for it to be done.