A "NIC" is a "network interface card" (or, often these days, "network interface controller"). It's set of chips (these days, single chip) which implements the technology for the network interface. An "ethernet controller" will take a short queue of packets and implement DMAing them from RAM, framing the packets (adding the source address and ethernet CRC). A modern ethernet controller will also be able to do some Network and Transport layer functions so that the CPU can handle packets more efficiently -- these "offload" functions include calculating checksums and packetising TCP segments. The controller might also implement the more physical aspects of the ethernet protocol, such as Base-T's autonegotiation, or it might leave this to another chip we call a "PHY".
The "ethernet contoller" for a switch takes this to extremes, implementing the entire "transparent ethernet bridging" forwarding plane between 4 to 16 ports. Recent controllers will handle most of the ethernet bridging control plane too.
In almost all modern router designs the router ports attach to one or more switch ethernet controllers, typically a Broadcom Trident- or Tomahawk-series controller. That controller might implement the router's forwarding plane, or it might simply handle port fan-out.
So, back in the IT Director's day routers would have a single ethernet port attached to a single ethernet network interface card, there might be four of these cards assembled with some routing logic to make a "line card". The "line card" would attach to the router's backplane.
Today even a low-end server NIC will happily present 4 gigabit ethernet ports. A switch or router might present 16 ports from the one ethernet controller. So your IT Director really should starting saying "port".
But before you go accuse your Director living in the past, consider that the job of a IT Director doesn't allow much time for the details of the technology; that's surprisingly low on the list of things they'd most like to know. So as much as they are being wrong in their terminology, bringing it to their attention in anything other than an aside, is to be wrong from the point of view of their expertise -- management. They have things to do with their time which are much more significant. I'd go no further than "just so you're not embarrassed in a presentation, they call them 'ports' these days".