In many countries in the Southern Hemisphere, including Australia, New Zealand and South Africa the winter solstice starts on the 1st of June and the official day of winter starts on June 21.
That's not a quotation, and it's inaccurate. The Winter Solstice doesn't "start on the 1st of June". The Winter Solstice is a specific astonomical point in time. Wikipedia says:
In many countries in the Southern Hemisphere,
including Australia [3][4], New Zealand and
South Africa, winter begins on 1 June and ends
on 31 August.
"Winter" is a political construction and can be declared to start on different dates as appropriate.
Please, if you write something that looks like a quotation, make sure it is a quotation.
It looks like the assumption is that other places start their seasons the same date as the US does.