My thoughts
The punto has two cooling flows. 1 goes out towards the heater and also back to the engine, without going to the heater, via the plastic splitter on the heater hoses and then back via the metal tube at the back of the manifold.
The other flow is via the radiator. If you are stuck in traffic the radiator fan should come on and off without you noticing the engine is getting hotter or colder.
If the heater circuit is clogged there will be insufficient flow thru the heater when the engine idles. If the pump is damaged there will be insufficient flow through the heater.
If you have not done it is a good idea to get some hose pipes and flush out all of the bits of the system in both directions till you get clean water