Create new unit tests of API annotation/tests/api/ to replace obsolete non-unit ones

[glowka]

Problem: Current are almost always querying URL, with a lot of data, checking a lot of things at a time.

What to do: Some tests are OK (rare), some need to be split in two, some in three. Do it in additive manner to avoid repeating same mistakes - create completely new and once new are in place, remove old ones.

Applicable to: all tests marked with # TODO: split this obsolete test to make it MORE UNIT

Related (but not duplicate task): #41