Penny Loretto About.com offers a useful starting set of types of internships .
Internships for credit [i.e. academic programme credit bearing internships]
Non profit internships [i.e. volunteering]
Service learning [Community service work where there are specific learning outcomes]
Co-operative education [usually distinguishable from internship by the length of service which is typically > 1 year and often are combined with ongoing learning through attending taught modules]
Externships [or work shadowing – very short term internships, usually from a day to several weeks only]
A further addition to this list is Pro-bono internships, where the student undertakes work of some duration, usually relatively short, free of charge but for personal gain. This gain could be the intention of trying to demonstrate to a prospective employer their value in anticipation of an employment offer outcome or for more general curriculum vitae massaging. [8 originally in an article by Paulette Thomas, Special news report about life on the job and trends taking shape there, Wall Street Journal May 28 1996].
Portland State University “Options for the final requirement internship or independent research project” notes (for internships in Gerontology)  takes the discipline related intern type in two directions, “career exploration” and “advanced internship”. In the “career exploration” form the internship is used to “deepen the student’s understanding of a particular specialization” whereas in the “advanced internship” form the internship is used to enable the student to “apply, test and further develop skills learned” in the academic program. These options are very programme linked and these may be more generally referred to as “career preparation” and “competence development”. ‘Career exploration’ suggests a higher level of uncertainty on the student’s part in relation to the choice of career in mind or as a personal initial foray into a particular specialisation. These sub-types are very closely programme linked in Portland’s case but, in general need not to be. A student is equally likely to make the choice of an internship in a specific company to explore their affinity to that industrial sector, that size of company or that academic discipline.
The Wikipedia entry for intern  differentiates between ‘work experience’ internships, primarily for skills development (common in the case of non-final year students) and ‘research or dissertation’ internships, primarily which serve as a vehicle to help the student complete their final year individual project.
Of general interest to the project is the legal status of the student intern in these different types of internship. Whether the intern is legally an employee, independent contractor or other has implications in terms of employment law, requirement to pay the intern (and how much) and tax (for both parties) .
Emerging from the above are a set of generic characteristics of internships, it might be useful to test, in any survey, whether the competences considered important for the student and the same for all types of internships.
Extrinsic reward: [Paid, pro-bono]
Academic reward: [Academic credit bearing]
Intrinsic reward: [Experience gained, New skills developed, CV massaging, Employment offer]
Duration: [Full academic year, Part academic year (as part of an individual project), very short (work shadowing, etc.), Summer vacation]
Future: [Employment offer, Desire to work in placement company, Desire to work in Industry sector, Desire to work in Large/Medium/Small sized organisation]
Typical responses to these being Yes/No, possibly with deeper exploration in some cases. There may be other options within each of the above sets.
The Higher Education Academy (UK)
Produces an “Industrial Work Placement Practice Guide” 
Higher Education Academy – Engineering Subject Centre
The Engineering Subject Centre, located within Loughborough University produced an employers guide to industrial placements .