That didn't seem to work I'm afriad, thinking about it though, could it be that I need to somehow convert

$caseInfo->Contact = $data['Contact']; 

into an integer for it to be accepted by the database?

Well I'm off now, but basically that is what I was saying at the beginning... but my concern with your function is that it is getting the right DataObject and performing the update on it... as always Debug::show (or to log file) is you friend debug EVERYTHING and you'll figure it out!
and to confirm I very much doubt DataObject get_by_id, setField or write is broken. I raised a bug on it once and was embarrassed to find I had a it all wrong... if I'd have spent a little more effort debugging...

as swaiba said, it would help to see your myFifth class or just its static $db setup. Because from the screenshot i assume that you set up the contact field as boolean, that would explain why always 1 is saved to the db since both '1' and '2' casted to boolean = true. in your particiular case i would use 'Yes' => 1, 'No' => 0, or change the field type to integer

Thank you both for your help.

Apening, you are exactly right, I had figured this out and as you suspected the column in the database was cast as a boolean rather than an integer. Works perfectly now!