I feel like I cut in your conversations, but, since I have a different opinion, please let me explain. When I'm talking about the time difference, I would use these words, 遅れている、進んでいる. So, my translation would be: 日曜日にサマータイムが始まったので、今、オーストラリアのいくつかの地域では日本より2時間遅れています。 Another way to say it would be: ・・・・・ので、今、日本はオーストラリアのいくつかの地域より2時間進んでいます。

日曜日にサマータイムが始まったので、今、オーストラリアのいくつかの地域では日本より２時間早くなっています。

I think the way of Selly-san would be better to avoid misunderstanding. Or saying オーストラリアのいくつかの地域と日本との時差は~2時間です is more specific. By the way, I would say "Japan is behind Australia": 日本はオーストラリアより2時間遅れて[おくれて]います。If the goal is at 8:00pm, when you finish the race, we are 2 hours later than you.

I realise I should have added, that because it is 8.00 here and 6.00 there it is later here than there, which is why I originally wrote 日本より二時間遅くなっています。meaning (I thought) two hours later than Japan. I am not sure whether my Japanese is wrong or we misunderstand each other about how we describe time differences!

Oh, how interesting! In English, we would say that Australia is ahead of Japan (time wise), because it is already 8.00 here, and it is still 6.00 there - so we are ahead of you (like running a race, we are already up to 8.00 and you are still at 6.00 so we are in front).. Conversely, we would say that Japan is behind Australia (time wise) - is it the other way around in Japanese?