Tell that to the Star Trek: TNG writers.
SCOTT
One of them... I actually served
on two. This was the first...
she was also the first ship I ever
served on as Chief Engineer.
MAXWELL
How are you? I had no idea you
were on the Enterprise.
(turning, to Riker)
This was my Tactical Officer on
the Rutledge. Best I ever had.
And so on....
Did you not see where I already said that scripts are not meant to be seen by the audience, so their use of capitalization is meaningless? Scriptwriters often have little or no experience with prose writing, and nobody's expected to see their text except the cast, crew, and executives, so their use of spelling, punctuation, and style is often far looser than anything you'd see in a professionally copyedited prose work. A script is not a final work, but just a set of instructions for creating the actual work. So it's allowed to be rough.
More importantly, there is no precedent - zero! - for there to ever be two "Chief" engineers aboard any Trek ship or posting at any time in canon. Rather it has always been used exclusively to refer to one single individual who has that job, as the boss of that department.
Exclusively except in TNG's first season. We all know that. That's the entire point of this conversation -- to speculate about possible explanations for that anomaly.