go with a much shorter heel. the heel drags on the edge, causing the roughness. you should try a heel that's very short. maybe a half milimeter or so, i cant tell from the foto, but you may also be going too deep into the metal. not sure why it's a problem with copper and not in silver. i assume both are about the same hardness.
for push work, you always want your gravers to be absolutely sharp. graver control is much easier with sharp tools.
td lewis is right as is Monk. Its one of those things that hard to get a picture of. But basically what is happening is the point of the tool and the back end of the tool are in different places in the groove. So as the point is moving forward the back end has to swing wider to stay in the groove. This drives the back end of the tool into the wall of the cut, scraping it. The solution as Monk said is to shorten the heel, a lot. Its to keep the front and the back of the tool as close as possible to being in the same place in the cut.
tdlewis also gives good advice. If you were using an air powered graver its a little more forgiving in turning the tool into the cut. But with push engraving its different. You should hold the hand holding the tool kind of firm once the tool is at the right cutting angle. Its also useful to place the thumb of the hand with the tool on the block to steady your hand so it doesn't wobble as the block is turned. Then you turn the block into the tool. Takes practice to get into the right position, but once there it kind of stays with you forever.
This is a wild guess but I think the difference between the copper and the silver plate is the difference in the softness of the metals. Copper is softer, easier to cut and you probably have a tendency to drop your hand closer to the metal, making the copper cuts have even more tearing. This is because the tool's rear is falling deeper in the cut. The silverplate being hard you keep the tool higher because if you keep it lower like the copper the point breaks to easily.
But the cuts you have made are bright and looks unmarked within the cut. That's a good thing. Learning to engrave is basically making mistakes and then figuring out how not to make them the next time. Then after you do something without mistakes, learning to do it even better next time. I don't think there are many engravers who have not learned through grim experience.
One other thing to consider is the angle of the tilt of the graver. To make a bright cut keep the angle of tilt of the graver low rather than a steep angle. A deep cut increases the chance for heel drag. However keeping a shallow angle will increase the chance for a slip. It takes practice but in my opinion a shallow angle will give better results and reduce the chance for heel drag. This was an after thought I thought I should add.