How do you call a concrete water catching thing under outdoor water tap?
I would call it a "basin".

A "catch basin" often refers to something else:  A box that pipe(s) drain into, and another pipe drains out of.  The bottom of the box is lower than the outlet pipe.  Sediment can settle in the catch basin, and not flow into the outlet pipe.

If you bought a shallow concrete sink basin or bird bath basin that drained into a hose, you could drain the basin with only a modest amount of work.  Route a hose from the basin's drain to a place you want to water.  (You could put the hose 4 - 12 inches underground, to improve æsthetics and avoid tripping hazards.)  At the end of the hose, attach drip irrigation tubing.  Unfortunately, Netafim-style emitters would not work in this application, because they need a 2 psi pressure differential to emit water.