From a math/compsci perspective, the most common definition is neither of these. Algorithm-3 is a deterministic process which terminates in finite time and yields some output. Thus an algorithm under definition three is essentially akin to a computable function.
I was writing sloppily without checking a reference. However, I did want to include randomized algorithms, online algorithms, and suchlike under definition 1.
Yes, I’m suggesting that most of the time when people use the term “algorithm” they don’t mean to include those so there’s a useful third notion of algorithm. I’m still not completely sure which of these the OP actually intended.
I was writing sloppily without checking a reference. However, I did want to include randomized algorithms, online algorithms, and suchlike under definition 1.
Yes, I’m suggesting that most of the time when people use the term “algorithm” they don’t mean to include those so there’s a useful third notion of algorithm. I’m still not completely sure which of these the OP actually intended.