Colin –
There is a huge raft of stainless-steel alloys developed for all sorts of different properties and applications, and their designations if not the alloys themselves vary a lot around the world.
You may well need more than just one country's standards tables to identify any of these alloys. The USA is only one country and favours its own specifications, so if it's not listed in the AINSI it may well be in the BS: EN, German DIN or other; but finding them is accordingly as complicated!
307 grades seem a bit odd though. It looks as if it's very much a welding grade, as the only references to it I found without spending half the morning, are as welding wire and rod.