Yes, your understanding is correct on the responsibilities of the two techniques.
One Gap vs Two Gap
The one-gap lineman is only responsible for the one gap in the offensive linemen and he typically lines up directly in this gap on the defensive line. The two-gap lineman is technically responsible for the gap on either side of the offensive lineman that he is lined up across from. The main purpose of the one-gap technique is for the defensive lineman to get past the offensive lineman and either sack the quarterback or tackle the ball carrier behind the line of scrimmage. On the other hand, the main purpose of the two-gap technique is for the defensive line to "clog" the line of scrimmage which allows for the more athletic linebackers to make plays on the outside of the play or gives them time to support in case of a run up the middle.
3-4 vs 4-3
However, not all 3-4 defenses exclusively use the two-gap technique and not all 4-3 defenses use the one-gap technique. Often 4-3 defenses can drop a single linemen into pass coverage leaving the other three defensive linemen playing a two-gap technique. Also, when playing the 3-4 and expecting a run play, defenses will often bring two linebackers to the defensive line just before the snap of the ball in support. In this case, all five defenders will play a one-gap technique, this is sometimes referred to as a 5-2 defense though there is not much difference between the 5-2 and the 3-4.
Explaining the Techinque
As far as having a better way to explain it to the player, I think the way you stated it is pretty good. However, it is important to note that with the one-gap technique the main objective is to get into the backfield and make a tackle or generally cause havoc. But with the two-gap technique, your main objective is to "clog" the line of scrimmage (draw double teams, prevent holes from opening, etc...) so that the linebackers can run free to the ball carrier and make the tackle. For this reason, linemen who play the two-gap technique are less likely to accrue lots of statistics for tackles, fumbles, etc... This is the main reason that many high-profile defensive linemen in the NFL do not like to play in 3-4 defensive schemes.