Tests: Make only integration tests rely on the WP core unit test infrastructure

#### Is your feature request related to a problem? Please describe.
Currently, most of the Jetpack tests (with the exception of the ones in /packages) run under the WP core unit test framework. Many of them aren't integration tests, so ideally they should run separately, just like the packages tests. Ideally, only integration tests should rely on the WP core unit test framework, and once that happens, we could run the integration tests only on master.

#### Describe the solution you'd like
Separate non-integration tests from the integration tests in Jetpack.

#### Describe alternatives you've considered
None

#### Additional context
Part of the Jetpack DNA project - p1HpG7-70O-p2

Start by locating Jetpack's test configuration and the places where tests depend on the WordPress core unit test framework; compare them with the tests under /packages. Identify which tests are integration tests and how they are invoked. Done means non-integration tests run separately, while only integration tests rely on the WordPress core unit test infrastructure.

Written by the indexing model from the issue text.