You are confusing two different operations dealing with Projections.
Define a projection: You would use this when you have a shapefile or other feature class that does not currently have a spatial reference defined when you look at the properties in ArcGIS. You might also use this if you knew based on some observation, that the projection defined for the layer was wrong. This could be due to recognizing that the coordinates of the features were not appropriate for the currently defined projection, or some other clue. This operation simply overwrites the existing projection with a different one that you specify. This can also be done in ArcCatalog by changing the coordinate system in the "XY Coordinate System" tab of the feature class Properties.
Project a featureclass: This tool is used when you have a featureclass with a defined projection, and you want to convert it to a different projection. This will run the appropriate transformation to change the features to the coordinate space of the new projection.
In your question, you state that you have a shapefile that is currently in a
Geographic coordinate system or
Latitude/Longtude which has units of
Decimal Degrees. This means that you have coordinates in the range of
Y/Lat = 0 to 90 and
X/Lon = -180 - +180. When you bring this shapefile into ArcMap untouched, its projection will be recognized. If you use the measure tool set to miles, ArcMap will calculate the distance correctly.
You then used the Define projection tool, which doesn't convert the coordinates, but simply changes the projection assigned to the shapefile from
State Plane NAD 83 California Zone 6 Feet. This gives you a shapefile with units in feet. The features in the shapefile did not have their internal coordinates converted, however, so they are still in the range of
X = -180 to +180 and Y = 0 to 90. Running the Project tool using the same coordinate system doesn't actually accomplish anything because you have already set your shapefile to this projection with the Define tool. This is why when you use the Measure tool, the distance is dramatically smaller. Essentially, your all the features of your shapefile are now fitting in a distance of 360 feet horizontally and 90 feet vertically.
The proper procedure, to be performed on the original shapefile in the geographic coordinate system, is to use the Project Tool. For the
State Plane, NAD 1983 California Zone 6 Feet. The output should be in the proper location and give a correct distance when measured. It should also overlay correctly on any other layers, provided they have a projection defined and that your Data Frame in ArcMap has a projection defined.