I think that defining "induced drag" simply as the "drag due to lift" misses some important subtleties. You can have induced drag with zero lift. Conversely, you can have drag due to lift caused by mechanisms that have nothing to do with induced drag.

First case:
Suppose you had a finite-span wing generating lift at say 5 degrees angle of attack (AOA). I think most would agree that such a wing would experience an "induced drag" force. Now suppose you were to take an identical wing, turn it upside down and attach it to the original wing by a set of long struts such that it was at -5 degrees AOA (kind of a mirror image biplane). If the wings were set far enough apart, both wings would generate roughly the same lift and drag as the original wing. The net lift would be zero (because the wings are lifting opposite each other), but the net drag would not. This is an example of induced drag with zero lift.

This is a silly example, but consider a wing with twist. There is some angle of attack where a twisted wing will generate zero lift. At the zero lift condition, some parts of the wing are pushing up and others down. There is induced drag associated with the local pushing even though the net lift is zero. The induced drag of a twisted wing can actually get smaller as it starts to generate lift. This makes it hard to identify induced drag as the drag due to lift.

Second case:
If you look at any wing section in "Theory of Wing Sections", the section drag coefficient (Cd) depends on the section lift coefficient (Cl). In other words every wing section experiences drag due to lift. However, I think most would agree that a 2D wing section experiences zero induced drag. This 2D "drag due to lift" is what I would call "profile drag", or drag due to viscous losses. Although some may bookkeep this as induced drag, the underlying mechanisms are entirely different.