I remember Walter Cronkite’s “The 20th Century” program from the 1960’s - always makes it easy for me to remember that dates like 1960 happened in the 20th century.
Because most people don’t have a problem decoding “xxth Century?”
They’re perfectly interchangeable, except for the few for whom it isn’t, and frankly, we don’t believe in coddling the weak 'round here. 
But as your link points out, it was renamed The 21st Century in 1967. And the 1900s were definitely not part of that.
The Fox stuff is even weirder. 20th Century Fox remains the name of the movie studio, etc. But its parent company is 21st Century Fox (taking on that name just a couple years ago).
Using such names as a memory aid is not recommended.
I don’t understand why so many people have trouble with ordinal vs. cardinal numbers. After all, your first year of life occurs before you turn 1, not after.
And don’t get me started on the Century of the Fruitbat.
It’s worth noting that the opening graphic for that show depicted the calendar/odometer clicking over from 2000 to 2001.
Because then you get into arguments over whether that should be 1700s or 1700’s. Personally, I think it looks odd without the greengrocer’s apostrophe. (Heh - autocorrect underlines 1700s but not 1700’s.)
College students, huh? What are the admission requirements - matching socks?
I would and do say “noon 45” or “midnight 45”
No they wouldn’t. “The 1700s” can refer to either the first decade of the 18th century, or any of the years from 1700-1799. Both uses are correct.
Would you say “The Second World War occurred in the 1900’s”?
How hard can it be? We live in the 21st century. The 21st-century’s years begin with 20. The 20th century’s years begin with 19.
It’s turtles all the way back.
Or go the other way. The first year of the “1st century” is (what we would now retroactively call) the year 1. The “first” hundred thus end with 100. The first year of the 2nd century is 101. The first year of the 3rd century is 201… et cetera.
It’s the mirror-imaged count in the BCE that takes a extra beat of thought to get right.
Jeopardy! often uses “the 1700s” to refer to the entire century.
1700s & 18th century are interchangeable.
99% of the time, it works every time.
At last! Somebody agrees with me. 
Yikes. I suppose you guys also want to start the forenoon watch at 8:01 a.m. instead of 8:00.
(BTW, that the C programming language starts with 0 instead of 1 has elegant consequences. When will you regular humans catch on? 
)
Times are stated like ages, in common English usage, indicated by the preceding completed term. Calendar years are like grades of school, numbered by the term in progress. (I understand that Chinese usage does ages of people in the latter way.) Alas for the logical inconsistency, but that is the deal.