Carboxylate "cage" concept
Let's start by deciding what compounds we're talking about here. You reference an example by your instructor of benzoate, which is the conjugate base of benzoic acid.



What your instructor is suggesting is correct for benzoate. The most important resonance structures for benzoate are the two equivalent structures that place negative charge on the oxygen atoms. It is also possible to draw a resonance structure where electrons are removed from the ring and both oxygens carry a negative charge.



The "carboxylate cage" essentially means that there is no way to delocalize the negative charge of the carboxylate into the aromatic ring. The only way to move electrons into the ring is from the C-O pi bond, which leads to (what I would consider) an unreasonable resonance structure, as it renders an oxygen both positively charged AND non-octet.



Carbonic acids are either a subset of carboxylic acids or a closely related functional group to carboxylic acids. Depending on what property you are interested in, they may be useful to compare or the difference might be too great. In this case, starting from the conjugate base of phenyl carbonic acid (as drawn in the question), there will be no way to delocalize the charge into the aromatic ring.