Hi Alan,
ArcMap has the ability to convert (project!) in-memory between different coordinate systems. To do so, it needs to know both the source and target coordinate systems. The layers or data are the sources. The target is the data frame's coordinate system. You need to leave the layer coordinate system definitions alone. They identify the current reference frame that the data is georeferenced to. If you want to create a copy of the data referenced to a different coordinate system, use the Project Tool.
The warning you're getting is specifically about having different geographic coordinate systems (also know as datums). Even if the data frame or a layer is georeferenced to a projected coordinate system, the projected coordinate system is based upon a geographic coordinate system.
One way to look at it--a geographic coordinate system identify the size and shape of the model of the Earth and where the latitude and longitude lines area. A projected coordinate system converts from this 3D system into a flat plane.
The same location on the ground will have different latitude-longitude values depending on which geographic coordinate system is used. The warning is that there are different geographic coordinate systems in use, and you may need to set some geographic (datum) transformations to reconcile them. The reason that we don't automatically apply geographic transformations is that there are often multiple ones per pair of geographic coordsys or a more accurate transformation may be to use two transformations that convert through a 3rd GCS! Starting at 10.1, we now present a sorted list of transformations in the data frame properties, coordinate system tab, transformation dialog.
Melita