A design patent application is limited to a single design. An inventor can file separate design patent applications for each unique design. But it is easy to see how this can quickly get out of control. To reduce costs, one approach is to file design patent applications for the actual design being marketed.
While minor variations may be considered additional embodiments of the same design, the Patent Office Examiners are typically fairly strict and will make the applicant select a single design for examination. Thus, while a single design patent application may include more than one embodiment of the same design, the Examiner may require the applicant to re-file these as separate design patent applications (and pay the separate fees for each).
But filing multiple design patent applications for minor variations of the same invention can also get the applicant into trouble. For example, this may be used as an admission by the applicant that the inventor considers such a minor changes to be distinct. A potential infringer could use this against the applicant later, countering any claim for infringement that any minor differences in their design also are distinct (and thus they are not infringing the design patent).
These are just some of reasons why it may be a good approach (when possible) to file both a design patent application for a specific design being marketed (regardless of how it functions), and a utility patent application for the utility of the invention (regardless of how it looks). Of course, how many and what type of patent applications to file is a business decision that should be based on input from a patent attorney who understands your invention and your goals for taking the product to market.