Thanks for writting this. First time seeing someone post something regarding these definitions and thought I’d comment on these (long post)…
I have seen many of these incubation and acceleration programs from all over the world and can tell you that the problem of a ‘correct’ definition (if there is one) lies in one thing: the type of Industry. If it is Technology startups (specially software), your first graphic fits nicely. While in traditional businesses, Acceleration seems to be done at a later stage in which the business already has a solid product and revenue (a typical case would be for when a company is doing business nationally and then wants to venture internationally (setup new factories there, open R&D units, expand sales offices, etc). The Acceleration phase would seem to fit well here as much needed capital and knowledge for expansion/growth would be needed).
Also, please note that on your second graphic you define the Incubator as ‘long-term’. I can tell you that in the case of tech startups there are many fixed term incubators (3–6–12 months max). That model gives a fair chance to all Entrepreneurs so they push their limits in order to get to the miminum viable product in a 3–6 month period after which they’ll be ready to pitch/launch their product. If they aren’t ready and they’ve ran out of time, then the resources that have been allocated to them (usually office space/mentoring time/marketing services etc) are reassigned to other Entrepreneurs.
Regarding your third graphic, there are also for profit incubators (taking either a small equity 1–3%, and some a little more provided they can help with seed investment at this stage). At the same time, there are some Accelerators that are non-profit (U.S. based MassChallenge comes to mind). It mostly depends on whether these institutions are tied to government or private sector (some tied to VC funds are for profit for example. And sometimes the VC firm even acts as the Accelerator in their own benefit), then there are those tied to Universities/Research Parks and those could either be for, or non-profit.
There is really a mix of things regarding these definitions so the only common denominator (if you don’t specify an Industry) is that the Acceleration phase comes after Incubation.