Fujitsu lifebook battery lifespan

So my classmate tells me that charging a fujitsu notebook battery while it's at full power will reduce its lifespan. He personally takes the battery out and runs it solely on the charger because of this. (also puts him at great risk of losing data because if someone steps on the charger and he hasn't saved, well.. Instantaneous shutdown) I was wondering if there was any truth to his statement; I did a search on google and I couldn't find any sources stating that plugging the charger into your notebook computer while your battery is at full power reduces battery lifespan. Does anyone know if this is true?

If it helps any, the label on the back of my battery case says 'li-ion 20' so I presume it is a lithium ion battery.

It is. Having your laptop charged at all times will wear out the battery worse than it will of you only charge it when it actually needs it.

My Fujitsu's battery died from bin plugged in for 14 days straight, after forgetting it before I went on vacation

It is true. Constant charging will damage the battery's life. Well, damage in the sense that you are not doing the most healthy thing which is to go through charge/discharge cycles regularly.

Even for Lithium ion batteries? I was told by someone else that smarter charging programs have solved that problem since the era of ni-cad batteries.

Also, will moving your laptop around while it's

1) in sleep mode
2) off
3) running

damage the parts inside of it?

Only thing in your laptop that's pretty delicate is the HDD's reader head, so I'd assume that it's safe in sleep and poweroff modes.