Please refer to the answer I provided here: What American football position does this phrase refer to: “He plays back for Dartmouth”?
The term fullback came from the positional verticality in the most common formations of the early days of football - the single wing and T. Your coach was right for the most part. The quarterback didn't even receive the hike in a lot of early formations.
When the quarterback went under center to receive the hike directly he kept the name because 1/16th-back didn't catch on (kidding, kind of). More importantly, at the time the terminology was evolving, the quarterback... wasn't important.
He was the guy who received the hike and handed the ball off. Quarterback wasn't the hero position it is today and for a matter of fact the QB getting the ball at snap created the "dead" era of offenses. Without the pass being a key factor the fact that the defense had a one man advantage lead to a lot of scoring problems.
That is why the fullback often got the ball during this era. The wildcat offense we know today is really just a variation of formations and plays from the 30-40s.
The fullback position evolved with the forward pass. Once the QB started becoming a weapon there was more of a need to get players out of the backfield and less need to "hide" running strategies. Just by aligning a player 2 yards in front of another player there is a whole other scheme you can use to attack defenses (Walsh being the epitome of execution in the West Coast offense).
So the fullback terminology really is about why did the fullback position of the T and Wing formations move from the furthest back to the "up" back in I type formations?
I don't think you can get an exact answer for this as there were formation shifts in the 50-60s due to the forward pass evolving and the terminology was different in different parts of the country. My guess on why the up-back became fullback is based off of the first iterations of "I" type schema.
If you take the wing formation from the 30-40s and then look at the power-I formation that sprung - this is a formation with a tradition I plus another lead back staggered next to the up-back (FB) - you will see that essentially they just crammed the wing formation into the backfield. Wing moves over into the staggered position, and half-back (this terminology wasn't used everywhere) and fullback moved into the other two positions.
Well in one iteration of the power-I the QB didn't receive the ball (he was taken out for a receiver or wing). The first up-back received the ball (pistol is innovative right?). Since the FB often received the ball on hike in the wing formations it was natural to put that same player there. You cannot hike the ball to a "fullback" if he is directly behind another player right?
The other thing that supports the move of the fullback to the fullest back to the up-back is that in the wing formation the fullback was generally your star player. When the power-I first evolved or like formations the "non-staggerred up-back" was generally used more than the back behind him. It was a "safer" play to jam it up the middle for 3-4 yards than to let the defense have another second to react to the runningback.
So the answer to where did fullback come from is really easy. It was part of the wing and T formation terminology. Now why did the term fullback evolve into a blocking back in the I formation? Well I gave you a really educated guess in that this same "star" player would logically move to the up position in the power-I and the term stuck. The fullback even in the power-I was a runner. During the 70s-80s the fullback position evolved into a lead-blocker role because the players spread out to defend the pass - and had to put smaller linebackers on the field - and this allowed the offense extra time to develop longer running plays.