How do coagulation profile tests work?

I work in veterinary medicine and we often need to run coag panels on pets to check for clotting factors.
We use blue top tubes with sodium citrate, fill them to the precise fill line, spin them down, then carefully siphon off the "citrated" plasma on the top layer.
I've been collecting samples for this test for years, and recently had an argument about the preparation method.
The doctor and an older tech told me they always sent the blue top tubes to the lab without spinning them down. The doctor thought that the lab needed rbcs or platelets to measure clotting time. I had to take out the lab's diagnostic manual to show him that the required sample was just the citrated plasma. However, I'm not a lab tech and I don't know how the test actually works.
Could someone explain why the sodium citrate is important for the test, why the plasma should be separated (and, if the test is negatively impacted by not separating it), and how the plasma is tested to get the clotting factor?
Also, one lab asks us to freeze the separated plasma before lab pick-up, and the other says to just refrigerate it. Is there a best storage method?