OK - Tired, sorry for mistakes on my part. Water pump didn't leak before these fixes so far so guess is it didn't hold pressure because of hose/radiator connections then pressure tested which left with engine running can exceed pressure (some types) blowing a seal in water pumps. No pressure with system intact really can exceed the pressure cap's rating, the tester will keep going way beyond that - forget for the moment if that.
It get hot quickly suggest it has air or vapor (combustion gasses from like a head gaskets are vapor) and still hot to touch but don't cool the engine.
Just looked and this water pump is timing chain driven and an alloy so subject to some corrosion. Impeller could be slow or impaired and would only know for sure looking at it and nuts not to replace pump if there. Pump looks like this........
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Part Number: BWP-9103Alternate Part Number: B 120-4190 Warranty: Limited Lifetime
Notes: This pump is driven off the timing chain. See vehicle manufacturer recommendation for replacement of timing components. *** Extended life antifreeze is highly recommended with this aluminum pump***.
The common first leak would be out a vent hole and could stop if fluid no longer there to leak or somehow at certain pressures?
I do think this will need a water pump but not convinced it's the up front issue yet but rather a head gasket putting combustion gasses into the cooling system. Nothing like overheating and despite this being older now it's still a mix of alloy metals and cast iron parts that REALLY don't like temp changes as they expand and contract at different rates hard on gaskets and alloys (include aluminum in that) cracks more easily.
What you might notice and bet pressure tester has been returned is fast pressure build up upon cold start up. You could also feel this in upper hose (only because it handy as pressure is throughout) first, then start up and pressure shouldn't be there so quickly but with combustion gasses would be long before normal expansion of coolant made pressure. You could EVERY SO CAREFULLY release pressure from pressure cap (WARNING WITH A WARM/HOT ENGINE FOR FLASH BOIL AND DON'T DO THIS IF NOT TOTALLY SURE OF YOURSELF AND WHAT YOU ARE DOING) and it would rebuild pressure again right away when cap installed again restarted, which is a strong clue the pressure isn't expansion but vapors - not what you want but I think that's the source problem now.
In short I'd do checks for head gasket problems on this while it's intact you can check more than if anything to do with cooling system is open or apart.
Sadly but they are common and messing with that radiator that ended up with wrong size hose diameter didn't help anything nor why it needed anything to begin with in its history.
Blaming a head gasket issue to me means ruling out as many other things that could fool that diagnosis. For this assorted tests of pressure, gasses in recovery tank, one or more spark plugs wet with coolant and more would be damning enough. Yup, they are costly and you don't know if heads are OK till they are checked out by a machine shop or plain seen when apart after testing is done about coolant because it's apart.
I can't know how much you want to dig into these things yourself or able to so think you should be seeking the right help from a tech and give him/her (yes gals into it too) know what the history is as you know it.
Both you and tech be up front with costs so nobody gets surprised if this gets costly or if time an alternative like stop now and sell as it is or choose something if bad enough or costly enough that you've had enough.
There's still a chance that it just isn't really full of coolant and free of air from draining and doing anything like radiator and hose things now thought a leak at water pump isn't looking like it's going to be all so simple now.
Get help if nothing more than a pro diagnosis and make moves from there as it's beating you up enough IMO.
Good luck, Tom