If you don't want to talk about the permutation group actions, just talk about "planar" or "nonpermutative" operads, which are probably more often called ["multicategories"](https://ncatlab.org/nlab/show/multicategory). They're what you get when you leave out everything about these permutation group actions. They may be good enough for what you're doing, or good enough to get started.
> Which law was this?
The second law relating permutations to composition. It's much less visually interesting than the law I included. Stasheff actually left it out of his book - the book I was reviewing! So, technically, almost all the theorems in his book are wrong.