The slides may have come from within NSA, but they were created for a non-technical audience of analysts. Suppose for the sake of argument that the reality is that the NSA has hardware adjacent to company servers or on the internet backbone nearby that can collect messages in transit that match some search query. That's not really direct access, and it's not indiscriminate -- all the subsequent statements from company execs and government sources would be correct. But it's easy to see how some mid-level worker in the NSA, trying to convey to analysts that PRISM enables them to retrieve emails, phone records etc. from various companies, would make an over-simplification and suggest that there is direct access to company servers.

"Never attribute to malice that which is adequately explained by stupidity." --Robert Hanlon. Despite the duplicitous and overly narrow statements about PRISM that skirt the truth, there's no reason to suggest they are all overtly lying, when a much simpler explanation is that some NSA employee overstated their technical capabilities on an internal powerpoint.