Hey there Martin
I trust by the user name Gooner that you're an Arsenal fan, if so good man
I was lucky enough to get a brief response from one of the developers at Bioware who are game company working on the very exciting Star Wars Old Republic MMO game. I asked about different areas of the production team and the dev pointed out you get all sorts doing different aspects of a certain job i.e. Modeller > creates the models. Animator > rigs and animates. Game Designers > designs the scenes and characters etc etc, Game Programmers > writes the programming to make the game work. Writers > Writes the contents of the game. The list goes on and Im sure some firms have a separate job for texturing models, sometimes they might want the modeller to do this as part of the package.
As you are swaying to the texture side of things. For this I would guess that you would need a decent enough understanding of the modeling process anyway though with ZBrush and Photoshop 3D (and many more I should imagine) applications, it's becoming easier to texture. I doubt the rigging is important for the texture though Im sure it helps.
But I reckon you should try and learn all areas of it though - modelling, texture, rigging and animation. Sure there is loads to learn and sure its hard, but no one ever said learning and hard work is not fun
I kinda see it all as going hand in hand because there are elements from each learning catogory that compliments the other.
Though I spend a good amount of time learning to develop each skill, I am a dabbler in Maya (fairly low skilled), Game development (extremely low skilled), comic book writer/artist (medium skilled). I tend to dabble and learn all these and develop each skill slowly because I don't really have a goal, just a passionate hobby of mine.
If you are still unsure, what I'd recommend is getting in touch with some companies that produce the kind of games you want to be involved in and see what they say. They might be able to help you define your goals.
Hope the above is helpful.