Regex to match full name with comma or space

I'm working with CommonMark and I've created a '@mention' parser that will return a link when an @mention is found. However, I'm tweaking it to link full names instead of usernames. Here's what I currently have:

$cursor->match('/^\w+,(.*)\s/');

However, this seems to capture the name incorrectly, and links the start of the other name (see below):

Does anyone have any ideas what I could change to match an entire name separated with a comma and/or space? I'm awful at writing regex.

Thanks for any help in advance!

Edit:

Here's what should match:

(A name can only be 2 words minimum and 3 words maximum)

• Bauman, Steve
• Steve Bauman
• Doe, Mary Sue
• Mary Sue Doe

A last name with a comma will only be one word.

What should not match:

• Steve
• Bauman
• Mary
• Doe

The regex needs to match an entire name, not single names.

This is the CommonMark code behind the match method:

if (!preg_match($regex, $subject, $matches, PREG_OFFSET_CAPTURE)) {
    return;
}

This results to:

if (!preg_match('/^\w+,(.*)\s/', "Bauman, Steve", $matches, PREG_OFFSET_CAPTURE)) {
    return;
}

As per your new rules, match a name that has 2 or 3 words and should be separated by spaces ( ) or commas (,)

/\w+([, ]+\w+){1,2}/

Regex 101