You need to widen your definition.
en·ter·prise [en-ter-prahyz] noun
- a project undertaken or to be undertaken, especially one that is important or difficult or that requires boldness or energy: To keep
the peace is a difficult enterprise.
- a plan for such a project.
- participation or engagement in such projects: Our country was formed by the enterprise of resolute men and women.
- boldness or readiness in undertaking; adventurous spirit; ingenuity.
- a company organized for commercial purposes; business firm.
As per the 5th definition that I have quoted above, Enterprise software or an Enterprise Licence is intended to provide a company-wide solution. As per the OP's dictionary definition, the "Project or Undertaking" in the cases quoted by the OP is a company.
A company-wide solution means that it will be used wherever a company has an office, or a site. So an Enterprise license will provide a solution for all of a company's sites. This differs to a Site license, which would only cover one location belonging to a company, instead of all of them. Anther use of the term Enterprise is simply to indicate an entire company, and isn't necessarily limited in definition to something software related. Thus you can have software, HR or Accounting, or even a supply of Jelly Beans for "The Enterprise". When used in either case, the word Enterprise in relation to the size of a company isn't relevant, as it applies equally to a small business, as it does to a megacorp.
Enterprise editions of software are intended to be packed full of the sort of features that you would typically need to suite a multi-site multi-server solution of some sort. In this case however, you'll find that it is just as likely to mean that you are expected to pay more to get more of the features that you probably wouldn't need in a smaller "enterprise", and therefore the word becomes as much marketing hyperbole as anything.
To answer the edited part of the OP's answer, an Enterprise Framework is a fancy name for an API that supports implementation of products designed for distributed data storage and communications, typically necessary for products designed with a multi-site product in mind. This is typically not intended for small businesses and yet can be used to create products which can scale from a single server/site to multiple servers/sites. The idea is to allow a product to grow to match the changing needs of a company (the Enterprise) as it grows, such as opening offices around the world, but sharing a common mail system for example.
So the issue of the word being used "vaguely" or "loosely" is really a bit of a non-issue, since the term Enterprise is always intended to mean "company-wide, regardless of the number of sites you need to service"... only Enterprise is clearly much less of a mouthful to say, and as per the examples that the OP has listed, the meaning of the word Enterprise remains clear.