Caller ID is the complete number (including country code) visible (or displayed) to the recipient.
Few countries and telephony service providers mask the caller ID in various situations, especially when received from an International destination or virtual number. When such a mask is applied, the recipient may see random (and incorrect or short) numbers instead of your actual full number. Not just that, the number may appear differently even if called to the same person from the same number again.
Most liberal countries and operators usually do not restrict Caller ID.
Caller ID service is variously known by many similar terms, such as CID, Calling Line Identification (CLI, CLID), Calling Number Delivery (CND), Calling Number Identification (CNID), Calling Line Identification Presentation (CLIP), and call display.