No, this is completely and utterly wrong. A swashbuckler is, if you want the iconic image, someone using light armor, a finesse weapon like a rapier and having a more carefree attitude. It is in no way tied to how many D6 you roll for damage.
I'm not 100% sure how to parse your sentence, but it seems to me that "a finesse weapon like a rapier" is within the cope of "using".
Read this way, you say that "a swashbuckler is . . . someone using . . . a rapier". What does "using" mean in this sentence? I posit that I don't turn my 5 year old child - who I can assure you is very charming and has a carefree attitue - into a swashbuckler simply by sticking a rapier into her hand. "Using" implies a certain competence, a capacity to use that rapier to genuine effect. And in D&D this is utterly tied to how many dice you roll for damage (plus your to hit bonus, your critical range, your static damage mods etc - if you want a summary expression, you could say it is wrapped up in your DPR).
For instance, if the typical duelist my swashbuckler will be facing has (say) 40 hp and deals 1d6+8 hp on a hit, and my swashbuckler PC has (say) 30 hp and deals 1d6+1 hp on a hit, then I have basically no chance of winning a duel with a typical duelist: assuming equal to hit chances, they will take me down in 3 blows, at which point I will have eliminated around one third of their hit points.
Under these conditions, within the fiction my character will be regarded - rightly - as a poor fencer, who is better than 5 year old children but nevertheless not very good with a sword. Certainly no one, either imaginary people in the fiction or real people at the gaming table, would look upon my character the way that we think of Robin Hood or Zorro.
At least at many gaming tables, and I think at those where D&D is being played in its default mode, the mechanics express the relative capabilities of characters. This is why a character concept cannot be divorced from its mechanical expression.