What’s the best way to cut the bottom of my door?
I'm not surprised that tore away from the edge of the work-piece.  Here are some ideas to get a better cut:
Get a fresh finish-cut saw blade.  The blade shown in your picture is well-worn.
Put something beneath the door while cutting it to avoid tear-out.  If you were using a table saw, the table itself would be helping, except this cut is so narrow it would depend on your table saw configuration.  Since you're using a circular saw instead, get a piece of scrap plywood to set the cut area on top of.  Set the saw blade depth so it cuts into the plywood slightly, but not all the way through it.