If my washing machine emailed me that would likely get lost in the noise. A Twitter DM is, relatively speaking, more noticeable.

(Although; an actual direct notification service would be even better)

Wait, really? You must use email and twitter in strange ways. I would also suggest setting up a "priority" email path for certain cases, like this.

On my phone there is no simple way to prioritise it; that happens when I sit in front of the computer and cut through my pile of email :)

So for something that I consistently need to notice at the time it arrives, things like SMS, Twitter DM or other low-traffic notifications are better.