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Does each court designate separately? For example, I have seen SCI designate β€œSenior Advocate”, and HCs, too, designate β€œSenior Advocate”. Is there are a difference between these designations?
I would answer all your queries after you read the India Jaisingh Judgment.
Swapnil Tripathi, Jaisingh etc. all of them have a common theme. The SCs grandiosity and progressiveness is restricted to judgements, the practice is something else altogether. The directions from both judgements I mentioned have not been effectuated for 3 odd years now. Hot take, but I feel that they (the puppeteers) think this divide is required in some darwinian sort of way, and no points for guessing why.
Any HC or the SC can designate a lawyer as "senior advocate", and a lawyer designated as a senior advocate by any HC or SC will be treated as a senior advocate of every court in India. So if lawyer A is designated by Bombay HC, he or she will be a senior advocate in all courts of India including the SC.
Is there any tangible benefits that one gets on being designated, other than optics?
No, optics matter a lot.

They help you draw more patience and also a fatter memo of fees.
Money. Lots of it. I work for one of the newly designated seniors in Delhi. Post designation he has doubled his fees. He used to make 1-1.5cr every year, he made that much last month. The quality of lawyering and the amount of work he puts into each case has remained the same, but the number of cases and amount he charges has drastically changed.