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It seems to me that the "disjunction gate" (aside: is that its proper name?) can be thought of as the combination of three gates, G1, G2, and G3, where G2 is the CCNOT gate, and $G1 = G3 = ¬_a \otimes ¬_b \otimes I_c$. For context, see the bottom two rows of the following figure that I created in draw.io:
Is there a name for the gate $¬_a \otimes ¬_b \otimes I_c$ (the one I'm calling NaNbIc)?
The gate in which you're interested:
would more often be called $X_1X_2I_3$ rather than $¬_a ¬_b I_c$, because we use $X$ to denote the NOT gate more often than we use $¬$.
However it is very unlikely that $X_1X_2I_3$ is given a new name, because there's nothing going on other than just two $X$ gates. CNOT has a name because the "C" part in CNOT denotes something that isn't quite so easily captured by something like $X,Y$ or $Z$. It seems that you have some familiarity with classical gates, is there any name for the 3-bit gate that does NOT to the first two bits and nothing to the third?
Welcome to QCSE. You seem to have gotten some of the gist of quantum gates but don't be surprised if not every such gate is promoted to having a specific, universally recognized name.
A reason some gates such as $\mathsf{CCNOT}$ (Toffoli) or $\mathsf{CSWAP}$ (Fredkin) have such a name is because they have been found to be useful and are a shorthand for the specific functions they perform. Indeed, the properties of both the $\mathsf{CCNOT}$ gate and the $\mathsf{CSWAP}$ gates were investigated in the context of reversible (classical) computing as studied in the 70's/early 80's, prior to/parallel to the beginnings of quantum computing.
Furthermore other lessons from classical computer engineering may form a context for quantum computing. For example, you properly identified a quantum $\mathsf{SWAP}$ gate, but classically promoting such an operation to a formal "gate" appears to be overkill. A classical computer engineer wishing to implement such a classical $\mathsf{SWAP}$ gate wouldn't likely consider any specific layout of CMOS transistors implementing a swap; rather, she would instead just consider jumping one input around another with a set of vias/metal runners. However, a quantum computer engineer (if there is such a term) would indeed be worried about how she implements the gate by means of the appropriate microwave/laser pulses, for example. A quantum $\mathsf{SWAP}$ is something entirely different from a classical swap; thus it's appropriate for us to give it a formal name.
Looking more at your specific gate, each input is a single-qubit gate, and there's no way for any entanglement to be created with such a gate. For example, if each of $\vert a\rangle$, $\vert b\rangle$, and $\vert c\rangle$ were unentangled prior to the operation of the $\mathsf{HOCKING}$ gate, they would be afterwards.
Thus don't be surprised if your gate doesn't have a specific, universally recognized name.
