Is there a name for this wave effect?

That's a pretty neat effect! It's not a bug in the simulation, it is correctly solving the equation.

To get a feel for what's going on, recall how a standing wave is formed. If you hold one end of a string and fix the other end, and give your end a wiggle, that wiggle will propagate down the string, bounce off the fixed end, return and bounce off the end you're holding, and so on. In practice, this would damp out quickly, but for an ideal string it would go on forever.

Now, if you continuously wiggle the string, the same process happens for each wiggle. After you've wiggled the string up and down $10$ times, there are $10$ wiggles continually bouncing back and forth.

If you're driven the string at one of the standing wave frequencies, then by the time the first wiggle comes back, you will be producing another with the same phase. They reinforce each other, producing a standing wave whose amplitude grows and grows. If you don't drive at a standing wave frequency, there will be a phase difference. For example, by the time your first wiggle has come back, you might be creating another one $90^\circ$ out of phase. The next will be $180^\circ$ out of phase, and the next $270^\circ$ out of phase, and all four of these will superpose to exactly zero, giving a stationary rope.

In general, this will happen whenever you drive at a rational multiple of the fundamental frequency. The reason the effect doesn't work when you change the tension setting is because that changes the wave speed and hence the frequencies, so you're no longer driving at a rational multiple of the fundamental. It doesn't violate conservation of energy, because for the last two you will be doing negative work on the rope. The effect may even be observable for simple multiples of the fundamental on a real string.

If you want to give this effect a name, it's simply the usual destructive interference, but with the neat twist that a wave you're putting in now is destructively interfering with a wave you put in earlier.