The only reason I would think it sounds the same is that the sustain pedal should sustain the grace note for just as long as the 2 whole noted, so it should have no reason to be different?
This is a fairly common idiom on the piano. The grace note falls on the beat - you are pedaling to sustain it so that all three notes are held at the same time, because the entire chord is beyond the left hand's span. You will probably get a slight "Scotch snap" effect, and that's expected. It is equivalent to, but less complex than, the following (but the composer of your example hasn't bothered to write in the lowest note with a tie, because the pedal is indicated and will have that effect anyway):
From the last movement of Paul Hindemith's 2nd Piano Sonata. Here both hands are doing the hopping (left and right both starting in the bass stave and moving to the treble), and Hindemith ties to the grace notes but doesn't bother to indicate pedaling, as it is pretty obvious where it is required.
You'll run across both styles of notating this in the literature, the second style with or without pedal indications (depending on how bad the composer's OCD is - it's something of an occupational hazard).
Arguably, the pedal would be applied after the grace note when notated like that. However, musically it makes more sense that this grace note is more of a "fingering" instruction, with the desired effect being what you spelled out in your second diagram, an on-time three-octave E though with a slight glissando and held via pedal.
However, if you try this exactly timed as written but without the final two notes, you'll find out that pedalling right after a short low note will have an interesting echo effect. Striking the final notes will mostly mask this, however. Playing around with such sound effects should really be done on an acoustic piano since it's anybody's guess just how much of those side effects a digital piano may or may not model.
Now for figuring out the indended effect, it might also be interesting to see what happens in the right hand at that time. If it is basically silent, the likelihood that the left hand is supposed to do some acoustic trickery is higher.
I would personally agree with user25636, that the composer wanted all three octaves in the final note, and realizing that you can't do that, set the bottom one in a grace note. If so, it follows that it should be held with the pedal.
The Ped. marking isn't terribly accurate in general, which is why the newer notation got some traction in the later 20th century. However, pedaling is generally open for a great deal of interpretation, so the markings should be considered guidelines. Play it as you like.