You had me curious about the rule after reading the thread. I looked up the official MLB rule book and this is the closest I found:
9.14 Base on Balls
A base on balls is defined in the Definition of Terms (Base on
Balls)
.
(a) The official scorer shall score a base on balls whenever a batter
is awarded first base because of four balls having been pitched
outside the strike zone, but when the fourth such ball touches
the batter it shall be scored as a “hit batter.”
Rule 9.14 (a ) Comment
: See Rule 9.16 (h) for the procedure
when more than one pitcher is involved in giving a base on
balls.
See also
Rule 9.15, which addresses situations in which
a substitute batter receives a base on balls.
(b) The official scorer shall score an intentional base on balls when
the pitcher makes no attempt to throw the last pitch to the bat-
ter into the strike zone, but purposely throws the ball wide to
the catcher outside the catcher’s box.
Rule 9.13(b) to 9.14(b)
(c) If a batter awarded a base on balls is called out for refusing to
advance to first base, the official scorer shall not credit the base
on balls and shall charge a time at bat.
(d) The official scorer shall score an intentional base on balls
when a batter is awarded first base because the defensive
team’s manager informs the umpire of the team’s intention to
walk the batter
So I am not sure if they are just basing the pitch hitting the batter being the "4th" pitch.