Yes, the only officially Activision-supported multiplayer version of Warcraft III at this time is the one with the webpage menu that has the "Warcraft III: Reforged" logo. You can play this version of the game without buying "Reforged" graphics, simply by entering your 2002 CD key into your Battle.net account.
However, there is no longer an officially sanctioned version of the Battle.net connection if you are using a version of the game with the old 3D menu background. Some people have created hacks that will change that old version so that it connects to a custom remake of the old server in the old style, but as far as I know any server that behaves in that manner is a 3rd party rewrite of the server and is not the original server, so if you connect to those 3rd party servers it is possible that a hashed version of your CD key may be transferred to the third party that is not Activision (as a part of what was originally meant to be the CD key verification process for the official servers).
Edit:
I am told that the old servers still exist for Diablo 2. The Diablo 2 remaster was made after the Warcraft 3 remaster and so by that time it was evident that deleting the old client created a risk that players might complain about losing access to a technology that they enjoyed. In this way, Diablo 2 and Warcraft 3 remasters are different in how they affect the old game.