A player concentrating on one spell casts another concentration spell ending concentration on the first, but gets counterspelled! A concentration spell that gets counterspelled does not require concentration. Is the player able to maintain the first concentration spell?

My initial response is, "no, you can't." You still cast the second spell, expending the slot and interrupting your concentration, but I'm wondering if anyone else has had this come up and if there's any Rule or Sage Advice that says otherwise.

Not you. >=P

Given that the ruling specifically effects you, I have to discount your input.

The rule states you lose concentration if:

Casting another spell that requires concentration. You lose concentration on a spell if you cast another spell that requires concentration. You can't concentrate on two spells at once.

I would say you have cast the spell if it can be countered otherwise why would you lose a spot? So I agree. 

Yes, you lose the concentration on the first spell when you begin casting the second.

If you want to look into semantics, look at the above for when you lose concentration on a spell ("Casting another spell that requires concentration") and compare it with the trigger for Counterspell itself: "1 reaction which you take when you see a creature within 60 feet of you casting a spell."

There's a distinct match in both wordings, right?

I would agree with the above that Counterspell requires a target of a spell being cast. Also, since the caster would have known they were casting a spell that would break concentration were it not to be counterspelled, I don't see this ruling as breaking anything.

Here it is straight from the horse's mouth:

"A concentration spell ends the instant you start casting another concentration spell, and a spell takes effect when its casting ends. If you cast invisibility in succession, you're visible during the 1-action casting time—effectively for a split second or so."