E.164 is the ITU-T standard that gives every phone number one unambiguous international form: a plus sign, a country calling code, and at most fifteen digits in total, with no spaces, brackets or dashes. +14155550142 is E.164; (415) 555-0142 is the same number written for a person.
What Spaw answers
e164 is the canonical form and the field to store. Three others carry the same number written differently: national for display inside its own country, international for display across borders, and rfc3966 as a tel: URI for a dial link.
Why the country parameter exists
E.164 is unambiguous only when the calling code is present. A number typed without one — 4155550142, or a national form beginning with a trunk 0 — could belong to several countries, so a lookup needs country to know which numbering plan to read it against. Without one the answer is missing_country, which is free.
An API key can carry a default country, applied only when the request body names none at all; an explicit country: null still means "read it from the calling code".