Greetings,
There are different ways each company handles it, but one common way is to jump straight to the root cause analysis mentioning only what triggered it (no need to double write things down). This and the CAR will of course be written in your own documents with the complaint attached to them (either paper or in my preferred way in pdf file and stored in your server/cloud etc).
It is useful to know the source of the NCR (internal/external) when you do your reviews/history checks, so there is no need to log it as internal.
Regards!