SwingsitlikeHogan
Major Champion
We don't need to ask the experts, you're being told the right thing which you could easily check!
Rule 10 - 1 b + c
After both players have started play of the hole, the ball farther from the hole is played first. If the balls are equidistant from the hole or their positions relative to the hole are not determinable, the ball to be played first should be decided by lot.
Slightly off on a tangent from the OP - but from the above - note that there is no mention of the green. So many seem to think that if they are 10ft away but off the green and I am 20ft away on the green - they play first because they are off the green. Oh well. I've never asked for a shot to be taken back if my opponent plays when closer, but off the green, and he hasn't asked me. One day I will.