With Clickatell it's easy to have the 'from' number of an SMS set to your organisation name - such as 'ABC NFP' rather than '01332 46000'. Several people have asked whether this can be the case for Twilio as well? (I'm asking this question because I know the answer! - see below)
Yes, it's easy. You have to ask Twilio customer support to enable 'Alphanumeric Sender ID' in your project. Here's the link for that. Once that's done then all you need to do is to set the 'From=' in Administer/Settings/SMS Providers to whatever you want. Of course, users cannot then reply or 'STOP' etc. But it works and there's nothing to enter or change in your Twilio account once the alphanumeric send ID is enabled, just update Civi with your chosen text.