1.0 player checking in! Might not have everything 100% accurate here, but I'm going off memory alone right now.
Bahamut ravaged most of Eorzea, if not all of it. People from all over the country definitely would have had a chance to see him, as the opening cinematic made abundantly clear. The chances of them knowing right then and there who he was is likely very slim, given that knowledge of what Dalamud contained was really only in the hands of a select few. That being said, people certainly knew of Bahamut, as there were NPCs that referenced him in 1.0. ("By Bahamut's Fang!" was a thing)
Word likely could have spread after the fact, though, and that would be that. What people aren't supposed to know is what happened to Bahamut after the fact. All the public knows is that he destroyed everything, and then simply vanished, never to be seen again. That is literally the end of the story, canonically, unless you play your character as the Warrior of Light and were the one who stopped his resurrection in ARR.
As far as the original Warriors of Light go, there's a pretty big discrepancy between them and the Warrior of Light. The Warriors of Light were present on the Carteneau Flats, and were a whole mess of adventurers who agreed to protect Archon Louisoix. Not all of these people were the Warrior of Light (The main character of 1.0, and subsequently of 2.0 and beyond), however all of them have been forgotten by most as of the calamity. As has been mentioned, anyone who tries to remember them only can recall a figure shrouded in light. They've been dubbed the "Twelvesblades" by the community, as far as I know. (I don't ever recall seeing them referenced this way in an official manner, unless it was offhand by Ferne in some post that I missed.)
Being a Twelvesblade does not mean having been an integral part of the 1.0 storyline. It simply means that they, as an adventurer, answered Louisoix's call to arms and were present on Carteneau, as a great many people were. The Archon saved these people in particular, however, as they did not receive the orders to retreat from the battlefield, and instead remained by his side.
From the perspective of the 1.0 main character, the "other" 1.0 players were these random adventurers that fought alongside them at Carteneau, but only they were the Warrior of Light in the 2.0 sense of the word.
Now, as far as memory goes, I can answer this question based off the ending of the legacy player storyline in ARR. I'll tag it with spoilers just because.
TL;DR
Yes, you can remember the calamity. Yes, you could have seen Bahamut destroy everything. You would not have known what happened to Bahamut after he suddenly vanished, nor would you remember of any adventurer friends you may have had that were saved by Louisoix (unless you have the echo).
Bahamut ravaged most of Eorzea, if not all of it. People from all over the country definitely would have had a chance to see him, as the opening cinematic made abundantly clear. The chances of them knowing right then and there who he was is likely very slim, given that knowledge of what Dalamud contained was really only in the hands of a select few. That being said, people certainly knew of Bahamut, as there were NPCs that referenced him in 1.0. ("By Bahamut's Fang!" was a thing)
Word likely could have spread after the fact, though, and that would be that. What people aren't supposed to know is what happened to Bahamut after the fact. All the public knows is that he destroyed everything, and then simply vanished, never to be seen again. That is literally the end of the story, canonically, unless you play your character as the Warrior of Light and were the one who stopped his resurrection in ARR.
As far as the original Warriors of Light go, there's a pretty big discrepancy between them and the Warrior of Light. The Warriors of Light were present on the Carteneau Flats, and were a whole mess of adventurers who agreed to protect Archon Louisoix. Not all of these people were the Warrior of Light (The main character of 1.0, and subsequently of 2.0 and beyond), however all of them have been forgotten by most as of the calamity. As has been mentioned, anyone who tries to remember them only can recall a figure shrouded in light. They've been dubbed the "Twelvesblades" by the community, as far as I know. (I don't ever recall seeing them referenced this way in an official manner, unless it was offhand by Ferne in some post that I missed.)
Being a Twelvesblade does not mean having been an integral part of the 1.0 storyline. It simply means that they, as an adventurer, answered Louisoix's call to arms and were present on Carteneau, as a great many people were. The Archon saved these people in particular, however, as they did not receive the orders to retreat from the battlefield, and instead remained by his side.
From the perspective of the 1.0 main character, the "other" 1.0 players were these random adventurers that fought alongside them at Carteneau, but only they were the Warrior of Light in the 2.0 sense of the word.
Now, as far as memory goes, I can answer this question based off the ending of the legacy player storyline in ARR. I'll tag it with spoilers just because.
TL;DR
Yes, you can remember the calamity. Yes, you could have seen Bahamut destroy everything. You would not have known what happened to Bahamut after he suddenly vanished, nor would you remember of any adventurer friends you may have had that were saved by Louisoix (unless you have the echo).
