You make a good point that some confusion may be resulting from different notions of algorithm.

Algorithm-1 is a technique that provably accomplishes some identifiable task X, perhaps with a known allowance for error. This is mostly what we talk about in computer science as "algorithms". Algorithm-2 is any process that can be described and that sometimes terminates. It might not come with any guarantees, it might not come with any clear objectives. A heuristic is an example of Algorithm-2.

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.