Not before
That's what nbf means.
The usage of this claim is optional and it identifies the time before which the token must not be accepted.
See the definition from the RFC 7519:
4.1.5. "nbf" (Not Before) Claim
The nbf (not before) claim identifies the time before which the
JWT MUST NOT be accepted for processing. The processing of the
nbf claim requires that the current date/time MUST be after or
equal to the not-before date/time listed in the nbf claim.
Implementers MAY provide for some small leeway, usually no more
than a few minutes, to account for clock skew. Its value MUST be a
number containing a NumericDate value. Use of this claim is
OPTIONAL.
The lifespan of the token starts after the time stated in the nbf claim and ends at the time stated in the exp claim.
Addressing your bonus question:
It's important to highlight that nbf is a standard claim registered in IANA while client_id is not. But it doesn't prevent client_id to be used.
Claim names can be defined at will by those using JWTs. The claims defined in the RFC 7519, however, are intentionally short:
All the names are short because a core goal of JWTs is for the representation to be compact.
This document also states the following:
Because a core goal of this specification is for the resulting representations to be compact, it is RECOMMENDED that the name be short -- that is, not to exceed 8 characters without a compelling reason to do so.