Effective nuclear charge and electron repulsion
Is the effective nuclear charge just a semiempirical value fitted to approximate the total potential energy of an electron in an orbital, or is there another meaning behind the number?


No, that's pretty much it. The Wikipedia entry has a partial periodic table with some $Z_\mathrm{eff}$ values. If you go further down to larger atoms, it becomes obvious that $Z_\mathrm{eff}$ decreases quite a lot towards the outer shells, so you could argue that the concept has its merits in capturing a quite significant effect.