I know that some games can only be played with the New 3DS, but then on others, like Hyrule Warriors Legends, it says, "Best enjoyed on New Nintendo 3DS". Keyword being best. So what I'm asking is, are some games being made as dual releases or do I have to buy a whip new system for 1 or 2 games?

As right now only two games are only for the New 3DS. Xenoblade Chronicles 3D and Binding of Issac: Rebirth. Some games like Majora's Mask make use of the extra c stick or run better like HW: Legends. 

Honestly you don't need to go out and buy the New 3DS unless you want a faster processor, better 3D, or new to the 3DS family.

Good, because we haven't even had our 3DS XL for a year. Thanks for answering!

No problem! Nintendo always releases many different versions of their handhelds every generation. It can get confusing.

I would assume "Best Enjoyed On New 3DS" implies something minor--the C-stick probably allows slightly easier control over the camera angles (eg Monster Hunter 4G) or maybe an easier way to charge attacks (eg SSB4). xD

Yep basically! Also the New 3DS has a built in amiibo reader. If you have the standard 3DS or 2DS they have a deceive you can buy for amiibo.

I have no Amiibo (or any intention of buying any) so that's not a problem either.

Any idea if there are any other New 3ds exclusive games planned?

Not sure to be honest. I highly doubt it at this point.