The simple answer is yes, they do. See Texas Rule of Civil Procedure 162.

But the defendant’s pleading must allege a cause of action independent of the plaintiff’s claims on which the defendant can recover.  If a defendant fails to do so, then the plaintiff has an absolute right to a non-suit of all claims, as the Dallas Court of Appeals demonstrated in In re Metropolitan Lloyds Insurance Company of Texas.

Metropolitan filed  a declaratory judgment seeking a declaration of its rights and obligations under the insured’s (Timberlake) insurance policy.  In her answer, Timberlake generally denied all allegations and then prayed that the court would enter judgment on her counterclaim against Metropolitan.  But she never named the cause of action nor explained the relief she sought.

Eventually, Metropolitan moved for non-suit without prejudice of all claims asserted in the lawsuit. The judge granted the non-suit.  A few days later, Timberlake filed a supplemental counterclaim which included new claims for various causes of action.

Metropolitan filed a motion for entry of a dismissal order, which the judge denied.  Metropolitan then sought mandamus relief.

Was generically mentioning "counterclaim" in her answer enough for Timberlake to have a valid counterclaim at the time Metropolitan non-suited?

No.  Although Timberlake’s answer contained the term "Counterclaim" in the title and mentioned it in the conclusion, her answer did nothing more than contest Metropolitan’s claims for relief.  As is well established in Texas law, any pleadings for affirmative relief must put the opposing party on notice of the nature and basic issues of the relief sought.  Timberlake failed to do that before Metropolitan filed the non-suit.  Thus, Timberlake’s counterclaim did not survive the non-suit.

For your reading pleasure, the opinion is located here.