I've had climber's elbow and tennis elbow for 10 years now. Is it still just tennis elbow or is it tendonosis?

Tendinosis. Tendinosis refers to a more chronic condition. You may want to see an orthopedic elbow specialist. Sometimes when conservative treatment fails, surgery may be required.
Same thing. Tennis elbow is also known as lateral epicondylitis and usually resolves with conservative management from 6mo to a year. If you have had symptoms for more than 1 year then you may need an evaluation and possibly an MRI to rule out lateral ligament injury or a complete tear of the insertion site of the extensors, there may also be some underlying joint pathology that could easily be ruled out.
Tendinosis. Studies have looked at the pathology of the tendon origins in tennis elbow and climbers elbow. Most of them have a tendinosis appearance without significant inflammatory cells. "tennis elbow" is just a way of describing the whole process.