You’re going to have to learn both, and then combine them. The two approaches you described are too restrictive and will sound boring after awhile. Also learn to arpeggio the left hand.
Ok. I didn’t want a bunch of piano teachers rolling over in their graves.
I can learn both approaches. Focus on the two handed chord first.
Then learn the traditional method later.
I’m not interested in doing a lot of fancy stuff. There are times I do need to accompany myself on piano while I’m learning to sing the song.
I play guitar and sing. I have to keep what I do reasonably simple and practice until I don’t need to think about the guitar. Vocals take a lot of concentration.
I’d guess singing and playing piano is similar. Unless someone is a really, really good keyboardist. Singing and playing doesn’t seem to bother Elton John or Billy Joel.
I’m not sure there is a right or wrong way, and it depends on the song and the arrangement. Some have the left hand playing the bass and chords together and the right hand playing a single melodic line, while others have the left hand playing a single bass line and the right hand combining melody and chords.
You wrote that you wanted to play chords and sing. If you’re doing that, then you’re probably not going to want to be playing the melody on the piano because you’ll be singing it. In this case, your right hand will play chords and left hand plays the bass, perhaps thickened with added fifths or octaves.
The more different ways of doing things the better, so you can adjust to whatever you think works best for the particular song you’re working on.
I’m still at the beginning stages with piano. There’s some good lessons on YouTube.
My vocal lessons required a keyboard for scales and to learn melodies. I’ve started learning chords and it’s helping a lot with my voice lessons. I don’t see myself ever playing piano in public. It is fun, especially with the keyboard’s built in rhythm tracks.
I regret now that I didn’t get piano lessons in childhood. It’s a valuable skill to have.