You've posted this on the Windows forum but I'll assume that the essentials are similar for a Mac.
First of all "I have also told iTunes in Preferences to look for the iTunes Media Folder location on my external hard drive" is not exactly what that setting does. The iTunes Media folder location setting in Edit > Preferences > Advanced determines where iTunes will put new media when added to the library. It will, though, determine what iTunes does when you use the File > Library > Organize Library... function, which will move any media that's outside the media folders into that structure.
What you want to have is the data on your external drive structured like this:

where the settings on Edit > Preferences > Advanced should be as follows:

- This is the location of your media files - on your case, Volumes/MacBook Air Backup Drive/iTunes/iTunes Media. For a "well formed" layout the iTunes Media folder should be contained in the same folder in which the iTunes database file (iTunes Library.itl) is stored.
- When checked, this ensures that media files are placed in a consistent folder structure where file and folder names are set based on the metadata in the iTunes library
- This ensures that all media you add to the library (irrespective of the method by which it is added) goes into the media folders defined in (1). The rules about adding media are not as simple as you might think ...

To answer your question: "When I consolidate files in "Organise Library"-does this get saved onto my *internal* hard drive, and if yes, does it take up space?" ...
The Consolidate Files operation will:
- do nothing to media files that are already in the standard folder layout/locations on your external drive
- move any media files that are on your hard drive but in locations outside the standard layout into those folders (potentially renaming files if you have the Keep iTunes Media folder organize option checked
- copy any media that is on a different drive to the standard folder layout/locations on your external drive.
Only the third of these cases will increase the amount of space being used by the library on your external drive. If you've structured and configured your library on the external drive, it will not make any changes to your internal drive. Turingtest2's user tip on Make a split library portable goes into more detail about the preferred structure of the library folders, necessary to having a complete library on an external drive, as well as the steps that may be necessary to bring your library into the right layout.