How did I break this desoldering pump?
You probably didn't do anything wrong.  It's just a cheap tool that broke.

What might help with that kind to soldering iron is to wait until it is cold, then hold the metal barrel while changing the tip.  The little screw holes in the plastic aren't all that strong.

That desoldering iron is underpowered for actually doing any work.  You generally need more heating power to desolder than to solder.  Typical soldering irons start at 25 watts (yes, there are lower rated irons) so the 30 watt is just barely able to remove things you can solder with a low rated iron.

I don't like the spring loaded pumps.  They kick when you trigger them, and the tip skitters across the board.  All too often, the skittering tip takes the solder pad right off the board.

I'm no friend of the expensive pumps, either.  

The best tool I've ever found for removing solder is this thing: