This started for referring to a new trend or fashion overtaking an old one. So for instance:
Comedy is the new Rock'n'Roll
The idea would be that things you might have previously associated with being a Rock'n'Roll musician (money, fame, fans, attention) are now things that you get with comedy. So rather than aspire to be a rock star and practice an instrument, one might consider studying comedy to have that kind of success.
It's a sort of funny way of saying that, and the kind of person who "tells you what's cool and what isn't" in this way is an easy target for mockery. So the construct has been picked up in ever-more-absurd ways, as people try and twist it further.
Imagine if you and your friends are trying to make a recipe, and it called for butter and you only have olive oil. You might make the substitution, excusing it by saying "Everyone knows olive oil is the new butter". This doesn't mean that you think butter is out of style and no one "cool" uses it any more--you're making a joke by acting as if you believed that.
At this point, most all usages are for humorous effect.
Short answer: The cuts are visible marks to what he or she is feeling inside.
The above usage seems more literal than some others.
My best guess is that to put a punctuation mark on something means
to bring to (visible) completion
or
to (visibly) complete
A sentence is not complete until a punctuation mark is "put on it".
A couple other uses (from Vocabulary dot com):
Evan Berry put a punctuation mark on the blowout by returning one of Tennessee’s four interceptions 100 yards for a TD in the closing seconds.
Notice this is talking about an exceptional running play that produced more points in the blowout game, "in the closing seconds".
The use you found regarding the basketball team is similar, since it was the last home game of their season.
It used to be the family dinner was a given; the punctuation mark at the end of each day, absolute and non-negotiable.
Notice at the end of each day. So we're in the realm of something that completes something.
Wagner's final shot of the match, a grunting effortful bouncer that clipped Ian Bell, was a worthy punctuation mark to a grand effort.
NB his final shot
The riot was symptomatic of the area’s sense of isolation and served as a punctuation mark on 10 years of restiveness in Miami.
All my examples come from the same vocabulary dot com link; I had to scroll through the examples. Some refer to 'literal' punctuation marks.
As far as the example with cutting,
It expresses emotional pain or feelings that I’m unable to put into words. It puts a punctuation mark on what I’m feeling on the inside!
Could it (cutting oneself) be thought of as somehow completing what the person is "feeling on the inside"?That's not too satisfactory. However, if you consider punctuation mark in the more literal sense, as visible marks, then this is perhaps what this usage is doing. The cuts themselves are visible marks on what he or she is feeling inside.
So, I'll just add this, to put a punctuation mark on my answer:
It’s now official what many people had predicted: former Alaska Governor Sarah Palin has ended her political tease — which helped her promote books and raise money for her political groups — and has formally announced that she is not going to run for the 2012 Republican nomination:
...
The decision from Ms. Palin on Wednesday placed a punctuation mark on the Republican presidential field. Her intentions were the biggest remaining question in the race. [Emphases mine]
(The Moderate Voice, quoting The New York Times).
Best Answer
A "hue and cry" was originally a loud shout after a criminal ("Stop Thief!") to alert others nearby to chase after them and catch them.
Now, it usually means a public ruckus about something that may or may not be criminal or illegitimate. There could be a "hue and cry" about the AC making the building too cold for example. It could even just be a loud gossip and pity party that doesn't really mean to be productive or solve the problem the members are complaining about.