What is the difference between “take to court,” “take the court,” and “take someone to court”?

Today’s (April 27) New York Times carried a caption,“Clippers Take to Court, but in Protest” and the lead-copy:

The players made a statement a day after racially charged remarks attributed to Donald Sterling, the Clippers’ owner, by wearing their shirts inside out before playing the Warriors.

in its home page.

But, the headline of the article was altered to “With Uproar Around Sterling, Clippers Take the Court.”

Is there any difference between take to court and take the court?

Why did NYT bother rewording the headline (though it may be better to ask directly to NYT)?

By the way, I found the following question and answer in wordhippo.com:

Q. What's another word for take to court?
A. Here's a list of words you may be looking for.

Verb. prosecute, sue, take legal action, press charges, file a suit, bring a claim.

Does the phrase, “take to court / take the court” in the above NYT article simply mean “appear in the court / join the game”, or “sue / bring a claim” by chance?

“Take to court” in a legal context must take an object, i.e. you take someone to court if you sue them.

The usage in the NYT caption is different. There is no object, and if the sentence wasn’t written in the telegraphic style of headlines, there would be a definite article as well. I did not find this use of “take to” in some online dictionary entries I consulted, but here are a few examples of how it is used. It means basically “to enter.”

Take to the court:

Autistic twins take to the court, and lead team to win (news headline – note that “the” is not omitted, even though some other modifiers are)

Senior basketball players from across the South Bay will be given one more chance to represent their schools as they take to the court tonight.

Take to the sky:

At the new Coachella, partyers take to the sky (LA Times headline of an article about taking a private plane to the music festival)

Take to the water:

Prince William and wife Catherine plan to test their sea legs Friday racing America's Cup yachts against one another on Auckland Harbour on day five of their New Zealand tour. The royal couple will visit Team New Zealand's harbourside headquarters then, weather permitting, take to the water on rival America's Cup boats.

“Take to the court” and “take the court” are synonymous here. The NYT may have decided to go with the latter because it sounds more neutral, less dramatic or formal (in some contexts at least.) Also, “take to” has other idiomatic meanings, such as to get used to something, which have the potential to cause confusion. But as you suggest, it’s really impossible to say exactly why the change was made.

"Take [someone] to court" means to engage them in a court of law. Court [6].

"Take [to] the court" means to go onto the basketball court, aka playing field. The playing area is a court in basketball, tennis, squash, etc. It is "take the field" in soccer, football, etc. Court [12].

These are two completely different things, though obviously with some allegorical relations (both are conflict, occurring in a defined space).

Also obviously, in headlines or other summary or casual usage words may be dropped from these usual formulations. E.g. "Bulls take court boldly" is a headline, and headlines are seldom grammatically correct sentences. (In fact, they often err on the side of incoherence.)