If we're going to switch from the English term to the grammar term here, I think we should use "init-declarator or member-declarator". We would also need to do something about conditions, which don't explicitly say that the variable is initialized from its brace-or-equal-initializer.

Maybe it would be better to say something like:

A declarator can be followed by an initializer that specifies an initial value for the entity being declared. [Note: In some contexts, only the brace-or-equal-initializer form of initializer is permitted by the grammar, but these rules still apply.]

Hmm. It might also be better to put some distance between the initialization rules, which generally have an initializer and some target entity, and the special case of initializing in a declaration. Perhaps this paragraph should start by talking about initialization in general, and then say that one of the places this happens is when a declarator is followed by an initializer (and other places include initialization of function parameters from arguments and initialization of return values).

Agreed. [dcl.init] started out at explaining the syntax for initialization, and is only ever so slowly morphing into a general description of all kinds of initialization, where the invoking syntax is secondary. The introductory paragraph follows that anti-pattern quite nicely.

A declarator can be followed by an initializer that specifies an initial value for the entity being declared.

This is description is correct, however I think we should emphasize on the 'initializer' which is the topic of this section [dcl.init]

If we look at the introductory description of [dcl.decl] p1

The init-declarator-list appearing in a declaration is a comma-separated sequence of
declarators, each of which can have an initializer.

It is easy to understand [dcl.init] was intended for "initializer".