For most sites, I think that unique visitors is more important than visits.
Like Joshak said, this could differ based on the business goals. However, I think that it is more typical to be curious about the size of an audience, such as:
- How many people watched a TV show?
- How many clients do you serve?
- How many prospects inquired about our products?
Also, another reason to suggest visitors is because it does tell a (fairly) complete story, e.g. "200 different people visited your site last month." In contrast, saying that there were "500 different visits" seems to not tell as complete a story, opening the door to more questions (and that might be good for you).
As far as the number of visits goes, that can be affected by other factors. For example, a product that has a long sales cycle could result in the same prospective buyer to make many visits to a site to view/review the product(s). Also, a product or service that is not easily understood (or not well communicated) may also result in more visits per prospect or per closed sale.
Agreed, it's not absolutely clear for all cases, but you asked for a recommendation and seem to want to keep it simple. For that reason, I suggest visitors.
Lastly, for what it's worth:
- Googling "web site visitors" shows 17MM
- Googling "web site visits" shows 7MM
Both queries contained quotes around the entire query phrase. This method has been proven to expose some spurious results, but mostly when the comparisons are supersets and subsets of one another.