Hematologist actually does something with the blood ( well these days pushes buttons on a machine) Phlebotomist just draws the blood. I also know at least one Hematologist who would know the Clair logo, I call her mom.
I was actually a pretty decent phlebotomist back when I was a baby HM (Hospital Corpsman - Navy speak for medic) as a lab tech in early 70's after my first tour as a USMC medic - and agree it makes a huge difference to have a good one when you're on the receiving end of the needle. But even good folks have off days and miss - hopefully those are few and far between for those of you needing blood draws.
Haven't done a peripheral stick in years (decades - damn I'm getting old) - now I do central lines (big central veins) but that's usually after no-one can get a peripheral stick or we just need access like right now in trauma.