Some great answers here already - I'd definitely recommend #2 or #3 - the separation is good conceptually but also in practice.
It can be hard to predict things like load and traffic patterns on an API and customers we see who serve the API independently have an easier time of provisioning and scaling. If you have to do that munged in with human web access patterns it's less easy. Also your API usage might end up scaling up a lot faster than your web client and then you can see where to direct your efforts.
Between #2 #3 it really depends on your goals - I'd agree that #2 is probably the future of webapps - but maybe you want something more straightforward if that channel is only going to be one of many!
