"I tried to figure out what to do, but without doing anything it suddenly worked anyway!
Why is this?"
Did you do the different tests as different users? Different users have different PATHs. root,su, and user all have different PATHs.
"What does what does // in /usr//bin: mean?"
It means that whoever wrote the scripts to set up your PATH made a mistake. The mistake is harmless.
Be prepared. Create a LifeBoat CD.