I've seen testers (Michael Bolton, mainly) make a distinction between "testing" (looking for defects using a variety of different methods) and "checking" (running a scenario and then verifying a set of facts which, if false, imply a defect in the product). Under these definitions, checking is a subset of testing.
Examples: Most automated regression tests would be "checking". Manual testing that doesn't go "off the rails" would be manual checking, but as soon as a tester leaves a script and starts investigating other behavior, they've started exploratory testing. Ad-hoc testing would not be checking. Fuzzing and stability testing would normally not be "checks", but automated exploratory tests.
I feel like there are a number of cases where people say, "Tests should have feature X", but then there are 10 exceptions to the rule. However, many of these statements seem to be truer and clearer when stated as "Checks should have feature X". For example, "Given the same inputs, automated checks should always produce the same outputs unless there is a change to the code." However, there are many tests (stability, performance, semi-random input, etc.) that would naturally have some level of variation in their outputs that are useful but do not meet this criteria. However, I also can see how this distinction could seem overly pedantic.
Concluding question, or, TL;DR: Is the idea of "checking" as a subset of testing a useful one?
P.S. Edits to link articles / sources for this concept will be appreciated, if anyone here knows what I'm talking about.
P.P.S. What I'm trying to get at is, 'Is a distinction between "testing" vs. "checking" like this useful', not 'Is it used?' I know it's not a commonly used concept, but it is a concept that people have been trying to introduce. Does the concept of "checking" to describe a subset of "testing" help with communication about tests and testing? Would it be a good thing to introduce it to tester / QA vocabulary?