Any information and commentary made will be drawn from peer-reviewed research that has been published in scientific research and sports journals. I've provided the links to these articles for you to read for yourself as well.
First, do we actually jump when weightlifting? In a way, yes. Now, we're not talking about jumping in the sense of elevating the feet off of the floor as high as we can. We're talking about jumping in regards to creating the vertical propulsion on the bar, or the "drive" portion of the jump.
Biomechanical studies have shown the second pull and jumping be similar and nearly identical. One study explained that "the angular displacement of the two movements follows the same time-sequence: from proximal to distal direction, the hip joint muscles are activated first, followed by the knee joint while the ankle is activated last." (1)
While weightlifters's feet don't elevate high off the floor like a vertical jump, the feet elevate just enough for them to quickly reposition laterally in order for the lifter to transition under the bar. This is largely because of the load on the bar and the ability to coordinate the timing of the hips to retract with speed. If a person, however, produced the same force into the ground as they would when lifting, they would likely elevate from the ground. Because of this, the second pull of the lifts is described as a "vertical jump with extra load."(1)
Weightlifters are known to be some of the best jumpers in the world. One study suggests that this may be attributable to the "high degree of explosiveness (in the sense of mechanical power generated) required for success in the sport, and qualitative and quantitative similarities between the lifting movements and the vertical jump."(2)
Is using the word "jump" a good cue? That's up to you as a coach. It must also be made clear, however, that a cue is used to get athletes to do something the coach wants. A cue does not describe what is mechanically happening.
Some coaches don't like to use "jump" as a cue because jumping forces lifters up on their toes too soon, as well as slows down the transition under the bar. It should be made clear that jumping does not create this issue necessarily. It is a mechanical and/or timing issue that is not being done correctly and needs to be addressed. To address coming up on the toes too early, a simple fix is to cue the athlete to keep the foot on the ground longer.
Conversely, you may find that your athlete's can easily relate to the cue "jump" because of their past or current athletic background (i.e. jumping athletes). Part of a coaches job is to make things relatable to their athletes. This is one way.
Now that covers the firs phase of the jump. What about the second part of the jump - the landing?
Jumping and weightlifting also involves impact phases. Weightlifters and athletes must have the ability to absorb the impact from the ground and from the bar. One particular study found that "the correspondence in kinematics between impact phases of jumping and weightlifting tasks suggests similar strategies are used to perform both types of activities." It went on to conclude that "weightlifting tasks share similar biomechanics to landing from a jump (specifically at the ankle knee and hip)..."(3)
So to sum things up, to say that we aren't or shouldn't be jumping in weightlifting is wrong. Research has shown not just in the propulsion phase, but also more recently, in the landing phase, the mechanics of both are essentially the same. If you took away the video feedback and just had the empirical data to review of both a snatch or clean and a vertical jump, you would be seeing the same data and feedback.
A weightlifter's errors of jumping forward because the heels are coming off the floor too soon or transitioning under the bar too slowly is because of mechanical and timing issues, not "jumping" issues.
When in doubt, just jump!
1) Biomechanical Analysis of Snatch Movement and Vertical Jump: Similarities and Differences
2) Propulsion Forces as a Function of Intensity for Weightlifting and Vertical Jumping
3) Characteristics of Lower Extremity Work During the Impact Phase of Jumping and Weightlifting