Lukas Novak in a comment writes,
It seems to me that the theory [the Millian theory of proper name] must fail as soon as its psychological implications are considered (those about beliefs are among them). In a judgement "Peter is wise" Peter must be somehow represented, not just linguistically but mentally. And since we are not omniscient, Peter-qua-represented will not equal Peter-qua-real ("warts and all"). In other words, there will have to be some conceptual content corresponding to "Peter" through which Peter will be represented; i.e. a "Sinn" or imperfect "Art der Gegebenheit" of Peter.
BV: I agree. The human mind is finite. So when I make a judgment about Peter, it cannot be Peter himself who is before my mind, Peter with all his properties. And yet something must be before my mind if I am to affirm that Peter is wise or even just to entertain the proposition that Peter is wise. Furthermore, this thinking reference or mental reference is prior to any linguistic reference. We can call this the primacy of the intentional over the linguistic. Chisholm championed it, but it is a controversial thesis. Now what it is that I have before my mind if it is not Peter himself?
Here very difficult questions arise. It seems we need some intermediary item to mediate the mind's commerce with the thing in reality. One vexing question is whether this intermediary item is or is not an ontological constituent of the infinitely-propertied thing in reality. If the intermediary item is a Fregean sense, then it is not such a constituent, but belongs in a third world (Third Reich?) of its own, a realm of Platonica, sealed off from the realm of primary reference (the first world) containing things like Peter. If the intermediary item is a Castanedan guise, then it is an ontological constituent of Peter.
Connected with this is the dispute whether Husserl's noema is something like Frege's Sinn.
I agree that "there will have to be some conceptual content corresponding to 'Peter' through which Peter will be represented."
This seems to me completely unrelated to the question of rigidity/non-rigidity of reference. It seems to me that all Kripke & Co. can (and do) prove is that names (normally) refer rigidly. But in my opinion rigidity/non-rigidity is not part of the semantics of an expression (Kripke's tacit assumption), but a way of its usage. Undeniably, you can use even a description rigidly, if you choose so. ("The president of the U.S. might very well not be a president" is perfectly meaningful and true, if "the president of the U.S." is meant to rigidly refer to whomever satisfies the description in the actual world.).
BV. Now you have lost me, Lukas. Suppose sense determines reference. And suppose the sense of 'Socrates' is specified by the definite description, 'the wisest Greek philosopher.' Used attributively as opposed to referentially (Donnellan), this definite description is non-rigid: it picks out different individuals in different possible worlds. So if the sense of 'Socrates' is given by 'the wisest Greek philosopher,' then the reference of 'Socrates' will be non-rigid. What then do you mean by "completely unrelated"?
But IMHO there is something true in the "mere label" intuition about names. I take names to have a dual role: First, they serve as imagined labels we use to mark individuals in order to be able to uniquely identify them. So far Kripke's intuitions are correct. But this role of a name is non-linguistic; in this role the name is not a sign but an imagined quasi-property of the individual. We could as well use real labels, real or imagined colours, numbers etc. Once an individual is named ("baptized"), we always have a descriptive content the one who (in this context) bears the name so-and-so uniquely representing that individual at our disposal. If we marked our individuals by means of colours, we would need a special linguistic item to represent such a description: the linguistic phrase "the one who (in this context) is marked by the colour so-and-so". But since we used words and not colours as our labels, we can use these very words as shorthands for such descriptions - and this is (usually) the other, properly linguistic role of proper names. Just like all other categorematic (extra-logical) terms, names in this role stand for a mental content, a "something-qua-mentally-represented" and in virtue of this can linguistically refer to the named individual. Note that this relation of "referring to" is distinct from (and conditioned by) the extra-linguistic relation of "naming" or "being a label of". This is why the theory is not circular (pace Kripke). Many names have this "minimal" meaning; but there are others, like "Jack the Ripper", that are shorthands for more substantial descriptions. But this does not preclude their capability to be used to refer rigidly - which, I would say, is the same thing as to supposit de re (in modal and other (hyper)intensional contexts). You need not expel the "reference-fixing descritpion" from the sphere of meaning in order to save the possibility of rigid reference.
BV: You are on to something important here. We need to distinguish the tagging/labeling function of names from their properly linguistic function. Suppose you and I each have a black cat and that the cats are practically indistinguishable. To tell them apart, to identify them, to refer to them, I put a red collar on mine and you put a blue collar on yours. The collars are tags or labels. As you point out, in effect, these collars are not signs of the cats, but something like properties of them or features of them. The collars by themselves have no semantic or referential function. The collars are, in themselves, senseless tags. The baptizing of a cat is the attaching of a collar. Corresponding to the physical act of my attaching a red collar to my cat is the sense expressed by the sentence, 'the cat with the red collar is Bill's cat.'
What makes the red collar signify Bill's cat cannot be the merely physical fact that the cat wears a red collar. We are brought back to intentionality and sense. A mind (my mind) must intend to mark my cat with a red collar, and to communicate this intention to Lukas I must use some such sentence as 'the cat with the red collar is my cat.'
The semantic function of a name cannot be exhausted by the object to which it refers since no physical item (whether a cat collar or sounds in the air or marks on paper) refers to anything. There has to be more to the semantics of a name than the object to which it refers. A name that actually names something cannot be a senseless tag.