TwoSix
Everyone's literal second-favorite poster
"Grasp" certainly carries that meaning. But I would say calling it a "beam" works slightly against it, as does the attack mechanic. Generally if it's emanating from the caster and being aimed at the target, I would see that as an attack roll; if it's creating an effect AT the target, that would generally be a save. Contrast scorching ray with fireball as the classic example. Or produce flame versus sacred flame.Hmmm.
I thought the word "grasp" suggested. . . well, grasping, holding onto and squeezing tight for a moment and thus the damage. The holding onto can thus be modified to pulling, pushing, etc by an invocation. Not sure I see the muddle you mention in this description, but I will think on it more. I definitely don't want to confuse players.
Now, if you're envisioning the spell as firing a flexible beam that is aimed at the target and then wraps around them, maybe a rename to "eldritch chain"?