Neurologist didn’t formally diagnose?

I have recently been to a neurologist about several issues I have been having. One of them is my tremor. I’m a 20yo female and I have had these shakes for probably the last decade of my life. It runs in the family. Dad, Aunt, Grandpa all have it. I really do think it is essential tremor because they all have been diagnosed and my symptoms are nearly identical with theirs.

I discussed the possibility of essential tremor with my neurologist. This doctor even increased my dosage of propranolol (prescribed for another issue) to help treat my tremors. I was having increased difficulty writing, and noticed within the past year or so my tremor has also spread to my head instead of just my hands. This medication change helped a lot, and we even discussed this at my last appointment.

This doctor did not put essential tremor on my medical chart as a diagnosis. I’m wondering if there was something I missed, because it seemed that he agreed at least somewhat that it was essential tremor? Is this something I need to push for? Honestly, my symptoms are fairly well managed with my medication, so does it matter if I don’t need to seek further treatment? Sometimes at the end of the day after taking my propranolol in the morning I notice my shaking is worse, but I have seen improvement in my ease of handwriting and other fine motor skills, as well as my peers asking less about my shaking. However I feel like it might be valuable to have on my chart? First of all as a reason for the increased propranolol dosage and like as to have a complete medical history? Thoughts welcome.