[Jim McDougall]

Well noted Glen. However, these were not circumstances in which guidelines or regulations were being followed overall, despite the fact that most of the men, regular citizens, were simply participating in a quasi military function.
While Travis had some sort of 'appointment' as an officer, like many of these ranks/titles were more 'honorary' than official.
As such, where officers typically had carte blanche as far as uniforms, and swords of course, he might have grabbed a sword from any number of forms that suited his 'fancy' (quite literally).

With this portrait of Bowie, it appears to be a silver hilt eagle head of the crested form, which like most of the earlier types of these were typically made in England and sold to US. This type crested eagle head seems to have been a type of pommel fashioned by Thomas Bate of Birmingham, but used by many of the cutlers there so might have been any number of these Birmingham makers.
These seem to be from the period c.1805 (Mowbray, 1988, p.96).

Inerestingly, an eagle head sword with Spanish motto blade was posted here last year, which with that being the case the sword was thought of course Mexican as they used the eagle element as well. The character of the sword suggests overall it is Mexican, perhaps later than the 1820 period (of the Bowie portrait) and of course the lack of knuckleguard. What is interesting is the character of the hilt, especially the crossguard which recalls the style of some Masonic skull and cross bones hilts. While Freemasonry was notably present in the US, it was as well in Mexico.

Returning to swords at the Alamo, while most of the defenders were civilian, the only 'military' there were essentially militia. The unit that Daniel Boone was with was comprised of a number of volunteers who were formed much as they had been during the War of 1812. In that same manner, the most noted unit at the Alamo were the New Orleans Greys (whose flag may have been actually the only flag flown there in the famed siege). This group was assembled in a meeting in an establishment in New Orleans where also was a storehouse of equipment left from the War of 1812. Many of the men were of course quite young and their fathers had fought in the War of 1812.

With young men, preparing for war, with the hubris and bravado, and recalling their fathers and their service in the earlier war, it seems possible as they outfitted themselves (even to the grey uniforms) that they might have taken swords....the Starr M1812 sabers made for dragoons and militias.
It is known that some of these were used in the 'Texas Revolution' so it seems possible some of these men might have had them at the Alamo as well.

As always, the presence of swords is absent from mention in the otherwise keenly detailed accounts and literature on the Alamo, the classic case of absence of evidence.

While a bit off track from the Spanish motto theme, and this eagle head is clearly not 'crested' as with Bowie's, the example with Spanish motto is of interest .

Just noticed Midelburgo's post...THANK YOU!
These large letters used for the Spanish motto along with the cosmological symbols popularly used on blades with Solingen origin suggesting the 'motto' was in use of course in latter 18th c. . Though rejected 'officially' by the Spanish court, these blades were solicited privately so available to cutlers not only in Spain and Mexico, but perhaps even in the US.
So does Bowie's sword have a Spanish motto blade? who knows" . ?