When I get a message saying a player will be out up to three weeks, but his player status says 11 days, is that the difference in my medical budget? Would every manager get the same 'out 3 weeks' notice, but their medical budget gives some kind of odds of it being shorter?

They tell you how long they can be out up to with that kind of injury. The length of time they are out of course depends on the severity of the injury as well as the players health and make up rating and your training and medical budget I believe.

I notice that every time I get a note with a title that says "out 3 weeks", when I click through it's always less.  And the message always starts with "it wasn't as bad as it seemed".

Medical budget determines how long they will take to recover.

It says he will be out "up to" 3 weeks.  The "up to" is the key phrase.  The correspondance you get in your inbox always gives you the a range of time he COULD be out, but once you open the email it tells you specifics.  Somewhere on the site there is a page that lists ALL the injuries in the game and the range of times a guy could be out.  Some have really wide ranges of time, such as 3-12 months.