I think you think the notion of "channel" is more important or fundamental than it actually is. "Channel" is a property of (some) particular diagrams, not of a Lagrangian or of an interaction.
The correct process is this. The Lagrangian tells you what vertices you can have and the basic Feynman rules, which you have correctly identified. Then when you want to know the amplitude for a particular process you draw all of the possible diagrams for that process using the vertices you have. Then you calculate the amplitude of each diagram using the Feynman rules derived from the Lagrangian add the amplitudes together.
In the specific case of four-body processes (like 2 particles scattering off each other into a 2 particle state) you may find that some of the diagrams have a term like $1/(s-m^2)$ or $1/(t-m^2)$. We call those diagrams the "s-channel" or "t-channel" diagrams. But again, you only find that out once you have drawn all the possible diagrams and the terminology only makes sense in the context of 4-body processes.
For example, if you are interested in two $\phi$ particles scattering off each other, $\phi\phi \rightarrow \phi\phi$. In this case there would be three tree-level diagrams corresponding to $s$, $t$, and $u$.
If you were interested instead in $\phi\phi \rightarrow \varphi \overline{\varphi}$ you would find only one diagram that would happen to be an $s$-channel diagram.
If you were interested in $\phi\varphi \rightarrow \phi\varphi$ you would again find only one diagram that would be $t$-channel (or $u$-channel depending on how you define your momenta).
