Hmm. I have dates with various people for various reasons, and use it to mean approximately the same thing as an appointment.
However, dictionary definitions of terms don't always cover all the shared cultural understandings of a term. If I were to tell people I'm dating my mother - we have regular dates, after all - it would give a very wrong impression of our relationship.
While in English-speaking company, I most often try to work around the whole term and substitute it with "I'm in a relationship with N.N" (which is silly, really - I have A relationship with every single person I know, even if it amounts to nothing more than a polite nod when I pass them by on the hallway) or "N.N is my girlfriend/boyfriend/lady friend/gentleman friend" (depending on age and gender - and in case you were wondering, I consider 26 to be the point when somebody graduates from gf/bf to a lady/gentleman status
The above terms mean for me that there are mutually acknowledged romantic feelings, a commitment to each other and shared future plans. That leaves out a myriad of other relationships which include sex and/or intimacy but lack the commitment component; NSA relationships, FWBs, fuckbuddies, romantic friends...