There is no right answer. You must decide for yourself how you want to communicate the function of your system to the reader of your schematic. I personally use generic connector symbols and some type of label [1] to connect them to the rest of the circuit. This connection is then what tells the reader what every pin of a connector is used for.
You can of course make a project specific connector symbol. This is the approach i use for high pin count connectors that are used to connect multiple PCBs.
If you want to standardize some connectors (example the power supply, a communication link, …) over multiple projects then it might pay to also make a specialized connector symbol for that purpose.
Note [1]: A label in the context of KiCad is a net-label. This is any object that controls the net name and which can therefore be used to connect two or more points without the need to draw wires. The different labels differ in scope (global, hierarchical, local). See Hierarchical or flat schematic design, what is best for me? (How to deal with multi page schematics?) (Appendix)
Further reading: