I really want to know in detail how the voltage rise during life of European HPS lamps causes damage to CWA ballasts. Plus, I also wonder if GE HPS lamps such as the LU400/40 are safe to use on CWA ballasts since these are export lamps?

The CWA vs series choke needs each different lamp construction, mainly related to the arc voltage management. As a consequence the US lamps tend to run at wrong power or cycle on a series choke and the European may exhibit thermal instability or runaway when at CWA.

Because of the strong positive thermal feedback due to the lamps being of saturated vapor.
Lamps for the US market are especially designed to hàve this thermal feedback suppressed (construction of the thermal coupling between amalgam reservoir and tge arc and the cooling of both).
But this is at first extra complexity so cost to the lamp, plus it increases the spread of the exact arc voltage over individual lamps. For the CWA that is of no problem, because the concept mainrains the current over really wide voltage ranges and still maintains large voltage reserve for zero cross reignition (the voltage across the capacitor adds on top of the OCV at that instant).
With European series choke the positive feedback is supressed by designing the choke so it slightly saturates at normal current, so once the arc voltage increases, the core saturation decreases, lowering the current. On the other hand this concept needs the arc voltage to be within rather tight limits, in order to maintain the choke at the desired operating point, which the US lamp construction is not able to maintain due to manufacturing tolerances.

From what I understand, should operating European high pressure sodium lamps cause long term damage to CWA ballasts such as in the case of operating a Philips SON-T PIA PLUS 400w on a North American 400w S51 CWA high pressure sodium ballast? As I looked at the 2008 GE catalog, it says that their export high pressure sodium lamps from 150w to 1000w for the European market are said to safely operate on their corresponding CWA ballasts.

I don't think there is any risk for the ballast, assuming hot restarting during more frequent lamp cycling does not cause extra wear (so maybe ignitor could be affected)
The problem with HPS on CWA is really "just" the arctube temperature stability, which may yield lamp overheating in some cases, and premature cycling.
Again, could be the higher power designs are generally more stable (due to the size generally easier to be made more accurately so less parameter spread, and due to the high power may need some heat management measures anyway, that make them more thermally stable).
So if it is officially rated (by a reputable maker), there is nothing to worry about, you may trust them the dependencies were verified and found compatible. For sure their development teams have way greater resource and information base to asses that than any of us here...
The "unknown brands" claiming something the reputable do not offer is a risky business to rely on. Because there use to be a lot of dishonest marketing involved. And if there is no high value reputable brand behind (to be easily lost from shady business practices), there is not much to keep the marketers away from such dishonest methods.