National highway does not have to be designated as such by the local government; that would mean that the definition would change from country to country.
National highway one of the three highest designation given to roads; and basically refers to those roads that span large distances and connect cities. So, without reviewing them myself, it sounds from your description that those are right.
Expressway and Freeway are technically higher, but a better way to think of it is that they are National Highways that are built to a higher standard. That is, all three serve the same purpose; they are high priority roads that span large distances and connect major cities; but Expressways and Freeways have additions like dual carriage ways and higher levels of controlled access in order to make them safer for higher speeds.
Explain to the mapper that we never set the road priority based on one aspect alone; as it sounds like he is concentrating on speed. We look at all of the aspects, and weigh each of them. The weights are not equal, and might be different for different priorities and areas.. In western Canada, the priorities often do not match the speeds in the guidelines simply because our laws here tend to set road speeds at a different priority than a lot of other countries. More important is what they connect to, the distances they span, etc. (Now when I say we don't look at one aspect, you must remember that you definitely have to have the right level of controlled access before they can be Expressway or Freeway. If you don't have that, then it is National Highway or below. But just being built to freeway standards is not enough to assign it Freeway priority, it must span large distances and connect cities just like the National Highway priority below it).