I'm not an expert and I'm weaker than you but I had some triceps tendon pain and biceps discomfort due to poor starting position on the press. I was keeping my elbows close to my body, almost touching my ribs in the starting position. When I corrected my starting position to the proper one - elbows in front of the bar and wrist in alignment my triceps's tendon is happy. From the video I can see that your starting position is not correct - your arms are under the bar and support the weight in the bottom position alone without the help of you chest.
Like I've already said I'm not an expert - I have three weeks on starting strength but I still think that you might benefit from this advice because the starting position with elbows slightly in front made the press a lot more enjoyable for me.