There are a LOT of ways to cast on. The techniques page here has videos of lots of different kinds and some opinions on their advantages and disadvantages. Tubular cast-on, for example, can take quite a long time and be very fiddly. On the other hand, it is very stretchy and looks GREAT on ribbing. I often use it, but for a jumper with many stitches I might not want to spend hours on it.
Any way will work to create a row that you can knit from. Here are some of the main points I think people consider when deciding what to use:
-is it difficult or time-consuming to do?
-is it stretchy enough for what you want?
-how tidy or pretty does the edge look?
-will it wear well?
-will it feel nice on the garment?
Long-tail caston aka double cast-on seems quite popular.
For any item you knit, these will differ in importance. For example, you probably don’t care if a scarf has a stretchy edge, but for a sock it needs to stretch a lot to get around the calf.
She probably taught you that way because it’s easiest and fast to learn. Nobody will tell you it’s better. Some people find it better for them, depending on how fussy they are, if they want to bother learning a new way, what they usually knit. Even she, if she is experienced at knitting, would tell you if you asked ‘it’s not better or worse, but it’s what I usually teach new knitters’. Regardless of whether she prefers it for herself and uses it in all her projects or not. It is quite easy and simple IMO and is probably what I would use when teaching someone to knit. I can’t think of a situation where it would be the wrong cast-on. If you get a cast-on that you are happy with and works for your item really anything is fine, even the simple backwards loop.
If you find later on that your cast-on is too tight or loose, casting on with a different size needle can help with that without you having to learn a new technique, but leave that for later.
What kind of items are you knitting? If squares, scarves, bags etc. there is very little reason to avoid any particular technique.
For now just use whatever technique you find easiect and prefer, I promise there won’t be any difference you are capable of noticing yet!!! (sorry) both of those techniques described could be done in such a way that they were stretchy or tight enough or would work for any item.
Regarding tension, that is just practice. The more you knit, the more even your tension will get. And you can practice to make your tension tighter or looser too if you want to.
The short version is don’t worry and do it however you like it will turn out fine.
Sarah