So does email. Email will be around in 5-10 years and is not tied to a company. Who knows what will happen to Twitter.

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.

What's wrong with SMS?

It can cost extra money in many places? Also, hooking it up to the mobile phone network requires more hardware or reliance on a SMS gateway.

Many providers provide email addresses for each number that causes an SMS message to be sent.

Some providers, I think you mean.