How has the supply increased in this scenario? What if, instead of minting the coin, you just tell people that you've done so?

The supply of money has only increased if you're able to spend the putative addition to the money supply.

Are you asking how supply works in my hypothetical, simplified example where the point I'm trying to make is that new supply in isolation doesn't matter - what you do with it does?

Or as you asking how it happens in the real-world example of Japan, where their supply increased from 400000B JPY to 1200000B JPY between 1990 and present, while everything remained the same price? And how this is seriously problematic in their economy?

What matters isn't what you do with "money"; it's what you can do with it.

But there are other things that affect the value of money. Tripling the money supply and seeing the value of money stay constant tells you that something else was pulling the value of the yen up at the same time that additional supply was pulling it down.

This is not true, though, haha.

Which is why the Austrian model, which only takes into account the former, is obviously and woefully incomplete - and has been rejected.