Rush was so on point with this that I actually thought he had quoted something that I had posted before.
I tend to call them Super-Detachment. You could also call them Voltron-Formations, since they're built out of other formations.
To address the differences:
A Detachment is a Force Organization Chart, where you are allowed to select a certain amount of units from different categories, for a Benefit. These will be listed in the codices, usually in the 'Forces of the XXX' page of the book.
Examples: Combine Arms Detachment. Ravenwing Strike Force. Nemesis Strike Force.
A Formation Detachment is a stricter sub-section of a Detachment. Instead of freely selecting models from the different categories, most or all of the units have already been selected for you, or you get limited choices (choose from three of the following). These are found in codices and publications with a special symbol: three skulls in a circle of curved lines.
Examples: Demi-Company Formation, Crimson Death Formation, The Eight.
A Super-Detachment (my term) is a Formation, as listed above, but instead of specific units, it is made of a limited selection of dedicated Formations. These Formation Detachments 'nest', which means that the individual Formation Detachments receive their benefits, PLUS a special benefit for the Super-Detachment and all of its components.
Examples: Decurion, Craftworld Warhost, Hunter Contingent
As to the last question, a Reclusiam Command Squad is a free-standing formation, and can't exist as part of another Deatchment also. The exception to this is the Super-Detachments, and it specifies in those detachments that normally a formation can't benefit from two sets of rules, but in these cases, they can. (the first one that comes to mind because I read recently was in Skitarii... They have a formation made out of 3 Battle Maniples, and they get the benefit of all three Battle Maniples PLUS the larger formation. This is the one Tyriks mentioned in his original post).
More on the original post: The Gladius Super-Detachment is very special. it is created by using a Demi Company, which keeps all of the rules for being a Demi Company, PLUS it gains the rules for being used as a Gladius (which requires you to have one of the Aux formation choices). If you were only going to run the Demi Company, with nothing else, you would get it's benefit, without the Gladius benefits.
If you wanted to use a CAD side by side a Gladius, you can do so, but they don't share rules, because they don't interact. And a Reclusiam Command Squad doesn't count as an HQ, because it doesn't have the HQ symbol on its datasheet, it has the Formation symbol (3 skulls in a circle) on its datasheet.