Variations in cooking method for Jasmine rice

Yes, those are two boiling methods.
One is using "right" amount of water that get absorbed by rice (that indicate rice being dobe) and second one is using excess amount of water. Both methods are valid although first one is much more water savvy.
But I have no idea why (and how) rinse cooked rice with boiling water. It's similar advice to using pasta water to help the sauce stick but properly cooked rice have enough surface on itself.

I have no idea how best to rinse with boiling water; I've never prepared rice that way.

That being said, there have been many studies of how much water to use with rice, and most have shown that the vessel that the rice is being cooked in is quite significant.

It mostly has to do with how much moisture evaporates as you're cooking, and so cooking larger batches of rice typically requires a lower ratio of water than a smaller batch of rice. Changing to a tighter fitting lid would also mean less evaporation loss.

And so the first set of instructions, in which you drain the rice after cooking is going to be more reliable for someone who is a novice at cooking rice.

But rice will also change some as it sits in storage. It's possible that you might have a brand that is not "new crop", and thus has dried more from when it had been picked than some other brand.

And there are differences between strains of rice. Although "jasmine rice" is a major category, there can be some subtle variations between strains, and between even the same strain growing in two different places. So this can account for differences in cooking recommendations, but the really special variations are going to be held aside and sold under a more specific name in a country that appreciates rice more (and thus will pay a higher premium for it)

You can also get differences in rice with how it's milled -- that is, the outer layer is scraped off to remove the bran layer that would make it white rice. This will dramatically affect cooking time, and may affect recommendations for how or when to rinse the rice. The 'fine bran' on the rice can give an off flavor, but newer milling techniques has created 'no rinse' rice that doesn't require trying to rinse it off before cooking.

And then we get into issues that different people prefer their rice cooked differently. Do you want a drier rice where every grain is individual, or do you want something stickier that clumps together?

So basically ... neither of the instructions are necessarily wrong, and neither one is necessarily better without knowing what type of result you're going for. Both actually do steam the rice some, as there's the bit of time on the second one where you leave it covered for a few minutes.

I would try to cooking rice according to the package instructions the first time, and then modify those instructions if the results aren't quite what I wanted. Unless you have a rice cooker, in which case, use the appliance's instructions the first time.