The practice of Pottying from Birth goes by several other names.
Most commonly on the web, it is referred to as Elimination Communication (or ec for short) - a phrase coined by Ingrid Bauer in her book 'Diaper Free!'. I avoid using this term when trying to explain our pottying strategy to other parents. I think it makes me (and therefore the way I raise my children) seem very weird indeed. If I'm trying to present this as natural, logical and applicable to everyone, that isn't a good starting point.
I prefer the term Natural Infant Hygiene (nih) but it doesn't give much of an idea of what it's all about.
Most descriptive for other parents is Infant Potty Training (ipt) - but that tends to imply that the 'training' aspect (and outcome) might be similar to that associated with 2 to 3 year olds (what I would call 'conventional potty training') and it simply isn't.
I've heard tell of people's grand parents calling it In Arms Pottying which is a pretty accurate description.
Then there's Nappy Free! which sounds wonderful but suggests a complete lack of nappies (or any other absorbent assistance) and is rather daunting.
So for me, Pottying from Birth or Infant Pottying it is.
Four weeks in and another wash load of muslins - not quite as 'nappy free' as the name might suggest!