a) After three years of speculation, James McAvoy has confirmed that he’s quietly tied the knot with partner Lisa Liberati.
b)
James McAvoy has confirmed that he secretly married his American partner, Lisa Liberati.
Those are two nice examples. In sentence a), the lack of comma shows that the appositive is restictive, which means that you can't remove the name 'Lisa Liberati'—it's an essential part of the phrase. In sentence b), the name 'Lisa Liberati' is non-essential information, which means that it can be removed without affecting the meaning.
I'd read an article I found on the New York Times website ... Here is an example from the article:
My son, John, is awesome. (If you have just one son.)
The commas show that the name 'John' is non-essential information and can therefore be removed without affecting the syntax or core sense of the sentence. If you do this, you end up with:
My son is awesome. What the writer of the article is claiming, then, is that the sentence subject
My son suggests that there is only one son. This is true to some extent, but it doesn't carry very far.
But withhold the comma if not unique:
My son John is awesome. (If you have more than one son.)
By contrast, in this case the name 'John' is essential information in that it defines which son is being referred to, and so it cannot be removed withot affecting the meaning. The idea is that if a speaker feels it necessary to define which son, the suggestion is that there is more than one. Again, this principle gets you only so far.
The only difference in the examples a) and b) is the possessive 'his' and that seems to have governed the usage of a comma.
The main difference between the examples is the meaning, where in one case the appositive is retrictive and in the other it's non-restrictive. The possessive word does not govern comma usage, the meaning does.
As I've said above, in restrictive appositive phrases that name and state relation, the possessive word is not customarily used. In non-restrictive phrases, the normal rules of using possessive words apply.
With the New York Times article in mind a) would suggest James may have been practising polygamy and has tied the knot with one of many of his partners?
No. It merely says that Sarah has a partner and that his name is James.
What the New York Times article writer says about uniqueness has some validity but not much. Whether we interpret how many sons or partners somebody has comes from a lot more than whether an appositive phrase is restictive or not. In the case of having a partner, it's our cultural awareness that tells us that Sarah has only one partner since this is what we'd expect based on how people called Sarah and James tend to live.
Imagine this line from the gossip column of a newspaper:
Hollywood star Ellen de Santos was spotted yesterday at Walmart with daughter Ruby.
From this, would you interpret that Ruby is one of multiple daughters? Possibly, but possibly not too.
How about this?:
Hollywood star Ellen de Santos was spotted yesterday at Walmart with her daughter.
Do you think it's clear from this that there is only one daughter? I don't think it is.
It's a restrictive appositive, where 'partner James' is used both to name and state the relation of James to her.
What would be the non-restrictive usage using partner James?
her partner, James
The comma shows that 'James' is non-restictive, and therefore is not an essential part of the phrase.