Are there different tests for Vitamin D & Vitamin D3? Specifically: Vitamin D3 (25 OH D) and 25 (OH) Vitamin D?

Yes. When there is a reason to test, testing for 25-OH-VitD2 level and for 25-OH-VitD3 level, plus adding the results to get the total vitamin D level, can help the doctor check the source of the vitamin deficiency and the treatment progress. There is debate over when it would be needed to do these tests. Some doctors feel that giving people more vitamin D would avoid deficiencies and avoid lab tests.
Vitamin D. Testing for vitamin D levels is superfluous for most folks. The testing is several fold more expensive than an over-the-counter bottle -- $10-20 depending on 1000, 2500, or 5000 units per capsule and 120 capsules per bottle. Your doctor has to have a good reason to order a $65 dollar vitamin D test.