As far as I understand it locality is the rejection of action-at-a-distance. By this I mean that in a given frame of reference at a given instant of time (in that reference frame), two physical objects can only interact if they are in physical contact. Lorentz invariance requires that this is the case in all frames of reference and so direct interactions between physical objects can only occur if they are located at coincident spacetime points.
If this is correct, would it then be correct to say that locality is enforced in QFT by requiring that spacelike separated fields commute, i.e. $$[\phi(x),\phi(y)]=0\qquad\text{for}\quad (x-y)^{2}<0$$ and then following the argument in the classical case that I gave above, lorentz invariance requires that interactions between fields are point-like? Do we require locality of interactions to ensure causality?
I've been quite confused with the conceptual notion of locality and thought that I'd finally sorted it in my head, but now I'm not so sure. I'd really appreciate some help in understanding the conceptual notion of locality and why we require interactions to be point-like?