a strategy internship may mean several things and not be necessarily related to strategy consulting. We actually just opened such a position in our startup, and in our case that would represent a business development position - support for the CEO for marketing and sales strategy decisions, plus support for the marketing and sales team with day to day activities. I would say most of the time a strategy internship should be interpreted as a business development one, unless otherwise specified. The activities may be similar to the one performed in strategy consulting internships (data analysis, presentations, market research..) with less or no client meetings, more focused projects and less working hours.
Hope this helps,