What would be the category that this type of playing falls under? I don't think it's called arpeggiation because that would be a single note at a time, as far as I know. Any ideas?
I'd call it "Broken Chords": (Wikipedia writes)
...broken chords play chord notes out of sequence or more than one note but less than the full chord simultaneously.
It could be called an arpeggio, but some define an arpeggio as a specific kind of broken chord where the notes are played one at a time and in either strictly ascending or descending order. The term broken chord is more general than arpeggio and includes arpeggios (in the strictest sense) as well as other chord elements played separately or together, as in the examples in the question.
I don't believe that there is a single term-of-art for this, particularly that would apply only to piano. I would describe them as ostinatos. If I were speaking to one of my piano students, I might refer to the first pattern very informally as a rock, an back-and-forth or a rocking motion (because that's what it feels like to play it). The second one I might call an Alberti-style ostinato or an arpeggiation.