[_JP_]

Is the original board the one with the Celeron and the newer one (and the actual) the one with the Pentium 4?

The Pentium 4 in older laptop were huge power consumers and heated a lot. This because they were roughly the same as their desktop counterparts.
So my guess is: either it needs the 135W (PFC or non-PFC) or that charger you mentioned is just plain rubbish and it's failing on you.
Getting cheap items for power supplies is the worse thing to do. It puts you entire system at risk. Insufficient power or abnormal supply can (or most likely will) damage the components and render the computer unusable or even reduce it to trash.

I can tell you, from personal experience, that cheap power supplies are never a good or wise investment. I've had cheap cell-phone chargers that went up in smoke after two uses, computer PSUs that couldn't handle 50% of it's maximum wattage and either the capacitors or the voltage regulators exploded and laptop chargers that almost melted with overheating due to poor electrical build/design/parts.

Get a good charger, but first try to borrow a 135W (it could be hard, mostly now there are only 120W max, usually there are 90W) and see if the current problem persists.