First of all, congratulations! Transcribing and playing by ear is a big step in developing your musicianship in general and getting it to the next level. They are incredible tools for improvising, composing, phrasing, analysis, and many more music dynamics. It can be overwhelming for the beginner, and the best start is to just start doing it, so you are already on a great path.
though some suggest not to write up the notation at all which seems a little confusing to me.
The people that are suggesting you to not write down what you hear are most likely suggesting that you analyze what is being played, using both your ear and your instrument, before attempting to transcribe it. Or probably they mean that you need to practice to play what you hear, before practicing actual transcription. Whatever the case, don't over-think it, both skills (transcribing and playing by ear) help each other out, so practicing both at the same time is a good idea, and not an uncommon approach at all.
These are just two methods I have encountered so far, and am I sure there are others
They are not two different methods to achieve the same thing, they are two different things that achieve different things, but both are very useful at implementing the other. Playing by ear is more about translating what you think or what you hear into a note in your instrument or your voice. Transcribing, in the other hand, involves translating what you think or hear into the music staff. One is more like talking, the other is more like writing, both are incredibly useful skills for musicians to develop!
If you can easily play by ear, you can easily find the correct pitch and rhythm to write down in the transcription. If you can easily transcribe, you are aware of the whole pitch and rhythmic pallete a specific musical framework offers you. They are both abstractions of the same thing.
I think you are already good to start! Use what you already know, start with very simple songs. Use your ear to analyze what is going on in the song, play it in your instrument or sing it, and then transcribe it. That's a very effective, common, approach to practicing both transcription and play by ear.
Aimee Nolte has a very good video on exactly this:
The concern of doing it this way is that I would develop bad finger habits whilst to try to locate the correct notes, but at the same this method seems more efficient in the long run as I am physically playing what I am hearing then writing it down - making strong active associations.
During these exercises, try your best to keep good technique, but don't focus too much about it. Focus on hearing, playing, and transcribing successfully. Ideally, you have some times in the day or week to focus on that.
There are many approaches to transcription, and you can learn a lot by listening to experienced musicians. What to do, what to avoid, what to focus on, what to improve, all kinds of stuff. There's a lot of info in youtube and the internet in general, here's another one from another youtuber I follow regarding transcription that I think can be useful.