This has been debated for nearly 17 years. It isn’t going to be resolved here and now.
I would point out, however, that the above passage contains a tautology. When the debate is over whether Syrio is Jaqen, it is not determinative that Syrio cannot be Jaqen because “The First Sword of Bravos does not run”.
While it may well be that the First Sword of Bravos does not run, that says nothing about whether a Faceless Man who is an Imposter and is NOT, in fact, the First Sword of Bravos, will run or not. We simply do not know.
There are other ways for Jaqen to end up in the Black Cells beneath King’s Landing and Meryn Trant choosing to put him there is only one possibility. There are a host of others. However it is that Jaqen got there, all of them ultimately depend upon Jaqen choosing to remain there. Why? That question is never answered, though we can hope that some day perhaps, it might be.
So why does Jaqen end up in the Black Cells, why did he choose to stay, and why did he not immediately leave the area when freed from his Wagon cell? We do not know and can only speculate.
There are many, many reasons for this. It all has the feel of some darker magical prophecy to it. The best information leads us to believe that Jaqen was there because he chose to be there, and he stayed in the wagon cell because he chose to, and he did not leave the Harrenhall area after being freed because he chose to stay.
As it all seems to point to Jaqen choosing to be there, the only real thing that is holding him back in the area is Arya. That lends some credence to his posing as Syrio Forel – because of Arya – as well.
If Jaqen chooses to be in the cells beneath King’s Landing, he chooses to remain in the wagon, and he chooses to remain near Harrenhall all so he can deliver the invitation to join the Faceless Men to Arya Stark (presumably for some fated/magical/prophetic reason) – then it isn’t stretching it to suggest he is Syrio Forel, either.
I have no doubt that GRRM initially intended Syrio to “die” in that fight. However, the attachment of readers to Syrio and the mystery of the identity of Jaqen creates many possibilities and side-roads for GRRM to explore. I think the honest answer is that up until AFFC, GRRM did not know the answer to whether Syrio was Jaqen or not (in the sense the GRRM never bothered to “decide” the issue).
As Jaqen does reappear in AFFC, there seems little doubt to me that GRRM has decided that issue. Whether we the reader ever find out is unknown, but my guess is that by the end, at the very least, we will get a strong hint one way or the other.
