Discussion Groups / English Usage / January 2004

"Non-defining" is the usual term.  (You'll also see
"non-restrictive.)  I've never seen or heard "indefining," and
"non-definable" indicates something that cannot be defined, not
something that does not define.

The words between commas in your second question form a participal
phrase. Some people would call it a participal clause; not I.
"Brushing" in this sentence is a present participle that modifies
the noun "husband."  Because a participle is one form of a verb, it
can take an object, here "her cheek."  "With his hand" is a
prepositional phrase modifying "brushing" (or, if you prefer,
modifying "brushing her cheek"; it does the same thing either way).
The entire phrase is non-defining.  I suppose the best label for it
is "non-defining participal phrase."  That's what I could call it.

What is important to understand is that this sort of phrase is
syntactically different from a relative clause, which is attached to
the rest of the sentence by a relative pronoun or a subordinating
conjunction.  Here the participle attaches itself to the noun, and
the remainder of the phrase is dependent on the participle.  Unlike
dependent clauses, this phrase has no grammatical subject, though it
does have an object ("her cheek").

English frequently uses dependent clauses and participal phrases
almost interchangeable.  Good editors can tell when one is better
than the other, but often either can serve in a given context.  Your
second sentence could be rewritten as "Her husband, as he was
brushing her cheek with his hand, kissed her passionately.  Now you
have a dependent (or subordinate; same thing, different label)
clause, connected by the conjunction "as", with subject "he" and
predicate "was brushing."  "Was brushing" is the progressive (or
continuous) aspect of the verb; "brushing" does not function
adjectivally in this sentence.

I hope this isn't too much detail.  The answer is not as simple as
you might think.

If she had two husbands, only one of whom was brushing her cheek with his
hand, then that would be a defining, a.k.a. restrictive, phrase (a
participial phrase, as Bob pointed out.) "Her husband" has him pretty well
defined by the time the phrase comes around. A phrase is defining, or
restrictive, when it's necessary for the identification of the person,
place, or thing it modifies, and that's why it doesn't take commas.