Looking into the Parish Registers on this website I found a baptism for Thomas Barnsley that states it's a translation not a transcription. Your site has his baptism 22.2.1680 and father is George. Baptism at the Parish Church Sheffield.
I've found a document on Ancestry that has been translated or transcribed showing Baptism 22.2.1679/80.
Can someone explain whether the calendar years back in those days were different and did not start in January and end in December please?
You're right, the calendar was different in the 1600s.
England was still using the Julian calendar, which literally dated back to when Julius Caesar ran the corner shop.
But the Julian calendar was slightly off, a bit out of sync with the solar year, and over the centuries it had drifted out of line with the seasons.
We now use the Gregorian calendar, which was organised in 1582 by Pope Gregory XIII. England refused to adopt it for a long time, because it was too Catholic or something.
New Year's day was Lady Day, 25 March, and not in January which seems weird now but that's how it was. Actually, in Scotland it was already January 1 and a lot of people in England felt that the year started in January, but officially it started in March.
From 1751 we caved in and changed to the new calendar (see the Calendar (New Style) Act 1750) because this was after the Act of Union and it was just too much hassle having different calendars in England and Scotland.
It meant bumping the calendar along by 11 days, which a lot of people hated and even rioted about (well, that's the story anyway, "calendar riots" with people shouting "give us back our 11 days").
It's also the reason that the UK tax year runs from April to April, because adjusting the tax year (apart from a couple of minor changes) was just too much hassle when everyone was already in the huff about the calendar change.