All browsers use the "Public Suffix List", at https://publicsuffix.org/
Besides TLDs, it lists all suffixes "under" which people can have their own domain (for free or not).
This is also called the "eTLD+1" policy, eTLD being "effective TLD", so not necessarily a real "Top" Level Domain, but any suffix that works effectively as a TLD, hence allowing registrations under it.
Is there one definition of what constitutes a "registrable domain" universally accepted?
Maybe not, but at least for now everything has converged on Mozilla curated Public Suffix List aka PSL.
Note that there was (and still is) various attempts to do the same thing, typically by storing in the DNS itself data about "relationships" between domains.
Latest attempt was DBOUND: https://datatracker.ietf.org/wg/dbound/about/
Its charter had:
Various Internet protocols and applications require some mechanism for
determining whether two domain names are related. The meaning of
"related" in this context is not a unitary concept. The DBOUND working
group will develop one or more solutions to this family of problems,
and will clarify the types of relations relevant.
The current way most of this is handled is via a list published at
publicsuffix.org (commonly known as the "Public Suffix List" or
"PSL"), and the general goal is to accommodate anything people are
using that for today. However, there are broadly speaking two use
patterns. The first is a "top ancestor organization" case. In this
case, the goal is to find a single superordinate name in the DNS tree
that can properly make assertions about the policies and procedures of
subordinate names. The second is to determine, given two different
names, whether they are governed by the same administrative authority.
The goal of the DBOUND working group is to develop a unified solution,
if possible, for determining organizational domain boundaries.
The working group was closed in April 2017, by failure of finding a real working solution (people agreed there is a problem, but the work seemed too big and hence no offered solutions were convincing people enough to make it a working group document and advance it as a standard).
There are still some work by individuals, like:
Is it precise and decidable in all cases?
Yes, if you use the PSL.
It is not perfect, because the PSL is humanly curated. So some entries are missing on it, and some entries in it do not even resolve anymore so should be cleaned. However the problem is on authorization: which party should be authorized to change which content on this list?
Do all browsers agree on what counts as a "registrable domain"?
Browsers use the PSL, yes.
You can find on https://publicsuffix.org/learn/ a (non-exhaustive) list of software using it, like libraries.
Note how it is used also:
- by the DMARC specification (for email filtering)
- by the CAB Forum requirements, to require CA to issue certificates only if they validate properly per the PSL
- in HTML5
- by Let's Encrypt to apply rate-limiting