It’s surprising how many people I have to explain that to.
It’s surprising to me that anyone would attempt to convince someone that the end of the 60’s is something other than 12/31/69 or that 12/31/1960 is not part of the sixties. That’s kinda what “the 60s” means.
Decade designators are different than centuries purely by how we define them. Centuries are counted. Decades are “named” based on their number in the tens place. Like it or not, that is the way it is. It is not a math problem. It is a communication method. If communicating with reasonable people is not your goal, you have lots of good options on how you define terms. Technically, every day starts and ends a decade, which is merely a ten year period.
There is a difference even in different ways of defining centuries. I’ll agree that June 1 1900 is in the nineteenth century. It is also true that it is part of the 1900s. To say otherwise would be silly.
I’m not at all surprised that you have to explain it to people. I’m equally unsurprised that they don’t argue with you about it. :)