This is a question about terminology, as I am clearly confused on the topic.
The Wikipedia page on higher-order logic defines it as follows:
Higher-order logic is the union of first-, second-, third-, …, nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply.
So this is worded vaguely. Taken at face value, it says that higher-order logic is the union of all logics up to some arbitrary $n$. This would imply that higher-order logic is not one thing, but that for each $n$ there is a higher-order logic, so that higher-order logic is sort of just a catch-all term for logic of some order > 2, or what have you.
But then it says that higher-order logic admits quantification over arbitrary-order predicates, which one could interpret as being the union of first-, second-, third-, ... order logic without stopping at any $n$, meaning quantifiers of all orders are included. However, this does not imply that there are "infinite-order" predicates that can simultaneously quantify over all finite-order predicates, just that you can have arbitrarily large finite-order predicates.
Then on this Stanford page, it says under "Higher-Order Logic" that $\omega$-order logic is "type theory," with continuation into the transfinite being conceivable.
Is the correct idea that higher-order logic has all quantifiers of finite order, each of which quantifiers over all predicates up to a certain order type, whereas type theory has infinite-order quantifiers enabling you to quantify over all finite-order predicates?