At least I think.
The first quote is in the 'Balls' section. The second is in the 'Mulligan' section.
"Putts are not counted as a hit to the ball."
"Mulligan putts will also degrade your ball by two hits."
If I take this as is, then a normal putt would not count as a hit, but one mulligan would count as two hits on the ball's durability.
I hope this is an error and mulligan putts actually do not cost two hits!