AlternativeProcedure -
do I put his/her name, and log the time as PIC under the column "Pilot In Command" instead of "Co-Pilot" Time?
Yes!
Let me clear this up :
When you fly and it's your sector, you log P1, or PIC or whatever it says in your book. In the remarks column, you write PICUS. When it's the other guys sector you log P2.