But the 24th birthday is the start of the 25th year, no? It means that the 24th year is now complete, so the next one starts. Like someone's first birthday, when they turn one year old, marks the end of their first year in life and the start of their second one. So the number of someone's birthday is the number of years they have behind them. That kind of phrasing is used rather frequently, so that may be where the confusion stems from.
I think we may be just from different semantics on that. For me, the 24th birthday is the start of the 24th year - the year you spend at age 24 might be a better way to phrase it. I can see it your way of course, it's just not from my frame of reference I guess. Like, I had my 37th b-day last month, that day marked the beginning of my 37 years walking this earth. It's almost like not counting the first 12 months of infancy honestly lol (like the counter starts at 0 and you build up to age 1 and so on). It's just the way I know it at least.