Lawyers and judges care about the overall effect of a thing, not the technical steps you take to make it happen. The overall effect is: Someone goes to your website, and plays Minecraft for free. Therefore you gave them the service of playing Minecraft for free. Which is illegal.

The fact that the file comes from Microsoft's server is one of those "technicalities" that you've heard about some people sometimes, but not usually, getting off on.

Microsoft will probably argue that the login prompt is a technological protection measure, which makes it blanket illegal to work around, due to the over-broad DMCA. That technicality is about as likely to work against you, as the "it's from their server so I didn't distribute it" technicality is to work for you...

Mandatory disclaimer: I am not a lawyer and this is not legal advice.