Veering and backing are TRENDS which must be defined either at constant height (or quasi-horizontal surface like a pressure surface, e.g. 850mb) or at a constant location (2D location...e.g. a city) and constant time. In other words, we can either vary time or vary height. Let's say we're going to discuss this in terms of a quasi-horizontal surface (e.g. the surface/ground), which means that we are varying time. Backing winds, then, refer to winds, at a particular location (say, Oklahoma City), which are changing in a counterclockwise direction. For example, winds at 11am are from the SSW, but at 1pm they are from the SSE.
We can also talk about veering and backing winds in terms of a particular/constant time and location but varying height. In this case, backing winds mean that the wind direction is changing in a counterclockwise direction WITH HEIGHT. To help avoid confusion, we typically say that winds are "veering with height" or "backing with height". This is the veering or backing wind profile (profile indicating constant time, varying height) that you hear about.
So, in summary:
+ Veering/backing winds (w/o reference to "with height" or "profile") usually refers to winds that are changing IN TIME at a fixed location and height.
+ Veering/backing of winds WITH HEIGHT implies winds that change in a clockwise/counterclockwise manner at increasing heights at a fixed time and ground location.
As a reminder, veering winds with height implies warm-air advection, while backing winds with height (or a backing wind profile) implies cold air advection. Typically, in the US, we want to see a veering low-level wind PROFILE (so, veering with height) with a backing surface wind tendency. It's also pretty common to see backing winds with height in the mid and upper-levels, which can be favorable since it implies cold-air advection in the mid and upper levels, which can increase instability.
