For example, if you have a Node server using Express and you write a Jasmine test to make sure POST /someroute gives you the JSON you expect, is that still considered unit testing? I know it doesn't seem to fit the strict definition of the smallest testable part of an application. So if not, is there a certain term for this type of testing of routes?
According to wikipedia? Sure:
In computer programming, unit testing is a software testing method by which individual units of source code, sets of one or more computer program modules together with associated control data, usage procedures, and operating procedures, are tested to determine whether they are fit for use.
In other words: anything you somehow do to test a program can be considered a unit test.
But of course, you are probably asking about unit tests in the more narrow meaning, as defined by art-of-unit-testing for example. One aspect there:
Runs in memory (no DB or File access, for example)
Or in other words: a true unit test tests a single unit in isolation.
Coming from there: I would not consider the described approach as unit testing:
In that sense: I think what you are talking about is much more a function/integration test than a true unit test.