In seeking the score for this piece, I came across 2 frequent editions: one that shows to use the pedal for half the bar, the other to use 2 bursts of pedal
Which pedaling is correct?
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The edition shown in the second video is correct insofar as it matches Chopin's hand-written copy.
The edition used in the first video I am not able to identify. It's not any of the editions currently on IMSLP, and several Google searches, including image searches for different parts of the score, did not turn it up.
This is the Peters Sämtliche Pianoforte-Werke, Band X: Berceuse, Barcarolle, etc. edited by Herrmann Scholtz, which can be found on IMSLP.
There is an autograph copy, available on IMSLP, that was used for the first German edition. It clearly shows pedaling at the half-measure.
Almost every edition on IMSLP follows this autograph. The variations are the French first edition — the piece's first publication, the German came next — and editions clearly based on the French edition. These are wrong, based on the evidence of the autograph.
The Video #2 edition — the Scholtz — correctly follows Chopin's autograph. The Video #1 edition, however, seems to follow neither the French nor German editions. The French edition has twice-per-measure pedaling in mm. 13–14, for example; whereas, the Video #1 version maintains the first-half-of-the-measure pedaling in those measures. The departure from the autograph/German editions is obvious from the first measure.
This leaves three possibilities.
The Video #1 edition is just sloppy and poorly written, based on the French edition.
The Video #1 edition is based on the French edition, but the editor figured that the French edition's departures from first-half-of-the-measure pedaling (e.g., mm. 13–14) were errors.
The edition is an editorial performance edition, reflecting not Chopin's "urtext" (which may or may not have been available to the editor), but rather reflecting how the editor felt the piece was best performed.