Ok so as far as i can tell, when you are following a career, you are limited to how many basic career skills you can train in, following that career. Once used up, you cannot train in any further career skills - you have to change careers to gain more. However, there doesnt seem to be a limit on non career skill advances. So suppose I am playing a trollslayer with just 1 skill advance. That means i have to choose between Athletics, Resilience, intimidate and weaponskill to train up. Oddly, though, if i was playing a career that lacks these career skills, such as an apprentice wizard, i could train in all four of these, at an increased cost obviously. It would take a little longer, but it seems such a character would have a much more rounded character that the trollslayer.

Have i got that right, or do non career advances also count for the 10 slots available?

Per the FAQ, the lines and boxes on the character sheet are intended as limits, so you can take at most two non-career advances while in a career and each can only cost at most 5 advances, so your wizard's apprentice would be limited to training just two of the skills you listed.

What Mac said