The smaller ornaments indicate an editorial addition. These additions are typically based on alternative sources or common practice, but which cannot be conclusively determined by the source material.
These are a variant marking for an appoggiatura from above (see #10 in the image below).
This is clarified further by the ...
Hindustani (North Indian) Ornaments1
Alankara = Ornaments
A major difference between Hindustani and Western ornaments is that Alankara are tied to raga, in the sense that how the ornament is executed and on which notes it is executed are characteristically defined by the raga. Western ornamentation is by contrast more general, the ornament being influenced ...
Yes, the ornament, known as a turn, is executed A-G-F#-G. Unless otherwise specified, ornaments are played within the given key signature.
For additional reassurance, of the editions on IMSLP that offer explicit notation (see below), all give A-G-F#-G. The Barry Cooper / ABRSM edition also gives the same notation.
Short answer: Play it how you like. Chopin died more than 100 years before this particular piece was published, so we may know even less about it than some of his other pieces. It was published and became well-known at a time when the tastes of performers were somewhat different than what we believe them to have been in Chopin's time.
This is the first ...
Why are both correct?
Maybe they're not.
In general, in everything, when someone tells you something, it pays to consider who is telling you that thing and why. Does the person have an agenda that might influence what they're saying or how they say it? What are the person's beliefs, and do you agree with those beliefs?
The same is true of music editors. ...
In addition to user48353's excellent answer, I'd like to add: We want to target our historical knowledge as narrowly as possible to the composer (or even the piece!). Don't settle for acquiring one rule for music 1600-1750, another for 1750-1825, etc. If the composer wrote about ornamentation, like CPE Bach, read that. If not, check contemporary treatises. ...
the harmony in this bar is Eb (respective to your earlier posting)
the note F is an appoggiatura resolving to the 3rd (G) - in common practice it was on the beat but - in pop and jazz it is ahead of the beat:
To collect some facts:
In baroque every player was prepared to supply his or her own ornaments. So they were either left out entirely, or reduced to a single + to suggest a place for some.
Especially for J. S: Bach this does not apply, however, since he provided a pretty detailed translation table (see Wikipedia, section baroque music or below)
Today few ...
Often, Baroque composers left out ornaments so these may be the choice made by each editor. In addition, ornamental notation wasn't standard so different editors may interpret an ornament differently.
In addition, Baroque key signatures for minor keys were not consistent. Often one flat was left out or even one sharp.
From "The ornamentation in the Fitzwilliam Virginal Book with an introductory study of contemporary practice":
The single- and double-stroke ornament signs appear at first glance to be indiscriminately scattered over the music without purpose. Research into their use reveals them to be employed systematically, besides being decorative elements ...