Lets give this word its true definition here, no shirking about.

Ok, "small", yes it does mean that, but only in certain specific circumstances. Take the following examples:

"I'll have a wee whisky" - Make mine a double and no, don't put any bloody ice in it. And since it's you that's paying make it a decent one, off the top shelf.
"I'm a wee bit concerned about our current progress on this project" - following the resignation of the entire programming team, the only decent manager we have in the company and the cleaning lady we are up shit creak without a paddle. There is no way in hell it's going to get done by the deadline and you may as well sell your stock options now.
"I'll get out of bed in a wee bit" - The bed is warm, it's 20 below outside and there is no way you are going to get me to move.
"I'm just nipping out for a wee pack of cigarettes, back soon" - It was nice knowing you, take good care of our children and I'll see you in 15 years.
"It's a wee bit chilly out today" - small children can be seen frozen like popsicles on the street, there has been no running water for the last 5 days due to the extremity of the temperature and I just lost the big toe on my right foot to frost-bite.

Although of course, "A wee wee" is still just a little bit of piss.