Like anything.... its all relative.
You know like old.... older.... oldest.
Is it when you have lotsa lines in your face ?? Maybe you walk slower or can't react quite as quickly as maybe 50 or 60 years ago. I know plenty of people that buy into this old age crap because of the number. That's it..... its a number. That doesn't mean you have to act feeble and shuffle around. Put some spring into that step and put a smile on your face and you'll always be young.
My neighbor and drinking buddy is 92 years old/young. He knows he's old, but he's on the go all the time. He knows when he reaches a certain point and doesn't overextend his limits. He goes to the gym three times a week for cardio and loves going to casinos. He even has a girlfriend in the last 6 months or so. Betty's 88 years old/young. They do all kinds of things together and can do almost anything today except maybe... well, I just don't ask.
As for 40 or 50 being over the hill..... just don't buy into it.
Haha...... listening to the latest of music or wearing the latest fashions.... I can't really see that. I'm me and I live like I want to. I don't like a lot of today's music.... mainly because the majority isn't what I consider music and I still haven't found the area of that crap that I like, yet. So, I do my own thing. I have a religion and I'm up on about 95% of the world's current events. That doesn't mean I have to embrace it. Taking on a complete new look to fit in is kinda sending mixed signals in my opinion.
I am where I'm at because of every decision or path I've chosen along my life's journey and I'm not at all embarrassed by any of those decisions, so why would I embrace something that isn't me or I believe in just because it's new ??
Does this mean you're going to get a facelift 'to fit in' because you don't want to be yourself anymore ??
One more thing.... you said being old in your sixties.... I hardly consider that old. I believe the new old age is after 90 some time. If you don't make it that far... you've just outlived or abused your body in the process of getting old.