It's very simple: #1 is the "cleanest" less cluttered design. This is where the world is currently moving so I am not surprised it is what is currently leading this poll. I also think LO could move into that direction speaking in more general terms of LOs UI.
I voted #4 but I'd prefer a plain white backgtound → as Adrien's comment which sums up well my feelings.
Мне понравился 1 вариант, поживей остальных, посовременней, что ли.
I think that #3 and #4 are the better designs for a number of reasons. But looking at the results I see that as on this point the #1 gets the most, #2 gets the next, #3 gets the next and #4 gets the least votes. Makes me wonder if that is an artifact of the arrangement of the choices. If #4 was on the top of the list would it get the most?
I'm not a fan of #1 because of the extra colors. Keep it simple.
I like the clean look of #2, but not a fan of the cubes. Do they have a meaning?
I prefer the white background of #3, and I like the hexagons but I'd prefer if they didn't interfere with the text. (something more like #2)
Not sure what the meaning or purpose of using triangles was, but I guess returning to that permanently is fine, but they should not interfere with the text and the background should be white, not pale yellow. It looks aged or dirty.
As for the shape used, I think if something was done with the hexagon to indicate the 6 apps that are part of the suite, it would be a nice tie-in. (Writer, Calc, Impress, Draw, Math, Base)
I would opt for clean, crisp, and very easy to read.