You are wasting your time by classifying tests. Instead of discerning what defines a test we'll hone in on tests to avoid. If a test is:
then delete the offending test right now.
For testing to work your test suites can't be grounds for noise pollution. Nor can they be a museum for specimens fit for dissection. Decide on what you want to guarantee and work to achieve that guarantee within contraint. Tests themselves are un-tested chunks of code. Tests that exhibit any of the characteristics listed above lose local reasoning and are, therefore, hard for a human to verify.
Slow and flaky tests mean you can't form a feedback loop with them. It means people will stop running the test suites to drive development. I often will chalk up work in CI for build bots to test and also test things locally at the same time, racing the two to get feedback as soon as possible. Tags and simple test names provide a handle to hone in on specific areas of functionality that can be verified as new features are added. Fast tests also mean people will add more tests and while a test suite might continue to increase in time needed to finish, it is arguably a point to break test suites up into new test suites and, possibly, separate libraries and programs that have their own test suites. Decomposition shows its beautiful face once again.
A non-deterministic (i.e. flaky) test may seem to sometimes provide a guarantee but the reality is much bleaker: a non-deterministic test tests nothing. I am not talking about tests that fail because of the occasional third-party service going down or network issue. I know you will be accordingly play-fighting with swords if that happens. What I am referring to is the situation where tests are known to occasionally but the reason is unclear. Is it configuration with a database? A third party library? Some state setup or internals of the subject of the test? Flaky tests are white noise. Devs start to ignore them and must waste time determining what is at fault if they are to ascertain if the test failure is because of something they should truly be concerned about or "just because".
It is also a waste of time when a new feature is birthed into the system only to lead a dev on a surgery process of fixing an array of tests that now fail. This is distinct from intentional changes: a test might need fixing because you are intentionally migrating away from some older behaviour into a new one and doing so in-place. But tests should have isolation: bringing in new functionality shouldn't necessarily mean overlap on older functionality and, therefore, older tests.
It's helpful to delete tests and see if you would passionately defend against their deletion in the process. If there is no passionate defense you will not likely miss them when they are gone. A giant wall of tests is also a giant wall of maintenance burden and there is only so much energy a group of persons can apply to maintaining something they don't care about whatsoever.
Tests and types provide a degree of confidence, one that allows us to assuredly tell others something is more likely to be correct, such that is to say it is aligned with some specification or set of requirements. Lacing your codebase with questions that can be quickly answered with a clear yes or no helps aid confidence. Debating if something is truly a unit test or integration test or whatever test is the equivalent of the art communities cliché of "but is it art?"; humorous but not useful. Along with foundations such as quality release and deployment engineering, operations, visibility into running systems, and so forth, pushing things out to production becomes trivial with time. I obviously and hand-waving away from the concern of scale here. Scale drastically impacts trust and confidence, but many organisations are still paving a path forward and charting new territory in this space to still make shipping code something sane. Whatever you take from the above, the most important aspect about any kind of testing is to make sure you are asking yourself one primary question when writing tests: What will you assert?