14th, that's not a key the 2nd Doctor is holding in that image; he did have a standard Yale-type key. It's his Stattenheim Remote Control for his TARDIS (that the Time Lords gave him - temporarily) in "The Two Doctors". That image is a grab from the end of the story when he's summoning his TARDIS. This was a new item in his arsenal. At the start of the story he was surprised (and angry) when he found it, and its receptacle, installed in his console, shouting into the air at the Time Lords for thinking that he'd "never flown solo!"
The story apparently takes place somewhere between his forced regeneration at the end of "The War Games" and the 3rd Doctor's appearance in "Spearheads From Space". He's in the TARDIS with Jamie and mentions having just left Zoe in her time zone, and has this one thing to do for the Time Lords and needs Jamie's help (or something like that).
The remote was obviously a temporary thing and the Time Lords removed it, and all knowledge of the remote from his memory when they turned him into the 3rd Doctor because, what he's doing with the remote above, is pulling it away from the 6th Doctor who tried to grab it from him while yelling "A Stattenheim Remote Control? Where did you get that? I've always wanted one of those." This indicates all knowledge of the remote was wiped from his memory because if it wasn't, and the remote hadn't been removed, the Doctor (6th) would remember having used it just before regenerating into his 3rd self. But since neither he, nor any of his selves from the 3rd onwards, had or remembered using this remote, it's a safe bet the Time Lords made sure he never knew he'd used it. (The Rani's Stattenheim in "The Mark Of The Rani" didn't even trigger his memory of it, so those Time Lords must have done a really good job of wiping it from his memory.)
It's all wibbly-wobbly, timey-wimey. But it's the more streamlined version of the Stattenheim Remote than the one the Rani had. (I think there's a thread about it on TARDIS Builders.)
"What's wrong with being childish?! I like being childish." -3rd Doctor, "Terror of the Autons"