Mil/Civ operations.
Whowhenwhy you are correct in that the phraseology example I used is not co-ordination. This is because a civilian ATCO would not require it as the traffic would be "known" once its intentions (not above 3000') had been passed.
If the situation were to be changed, and the other traffic was descending towards mine, then I would expect the other controller to say something like " Roger I'll descend on top with my xxxx, let me know if yours climbs". Response "Wilco". S/he (sorry Flower) could then descend to a level below FL110, making sufficient allowance for any pressure difference, say FL50, and then further once lateral separation was achieved.
What I'm saying may not be right under military procedures, but it is acceptable under civilian ones.
As an aside, I do find it worrying that a military ATCO will, and has to my knowledge, disregard co-ordination because the phrase "co-ordination agreed" has not been used. I believe that this is one matter that is under investigation following at least one airprox.
Changing the subject slightly, an ex-colleague, a former Squadron Leader at a fast jet base before getting his civilian licences, so with a fair bit of military experience, once said to me that he had never worked so hard until he came to our unit.