An algorithm is “a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer”

How does a giant pile of linear algebra not meet that definition?

Deterministic (ieee 754 floats), terminates on all inputs, correctness (produces loss < X on N training/test inputs)

At most you can argue that there isn't a useful bounded loss on every possible input, but it turns out that humans don't achieve useful bounded loss on identifying arbitrary sets of pixels as a cat or whatever, either. Most problems NNs are aimed at are qualitative or probabilistic where provable bounds are less useful than Nth-percentile performance on real-world data.

How do you define "algorithm"? I suspect it is a definition I would find somewhat unusual. Not to say that I strictly disagree, but only because to my mind "neural net" suggests something a bit more concrete than "algorithm", so I might instead say that an artificial neural net is an implementation of an algorithm, rather than or something like that.

But, to my mind, something of the form "Train a neural network with an architecture generally like [blah], with a training method+data like [bleh], and save the result. Then, when inputs are received, run them through the NN in such-and-such way." would constitute an algorithm.

NN is a very wide term applied in different contexts.

When a NN is trained, it produces a set of parameters that basically define an algorithm to do inference with: it's a very big one though.

We also call that a NN (the joy of natural language).

Running inference on a model certainly is a algorithm.