NO, L3, L4 and L5 are NOT "very nearly the same distance as the moon." (Did you not realize the diagram would not be to scale?)
L4 and L5 form the third point in an equilateral triangle whose other two points are the Sun and the Earth. Given that all three sides of an equilateral are the same, and that one is the distance from the Earth to the Sun (roughly 8 light minutes or 93 million miles, if memory serves), that means that the distance from the Earth to L4/L5 (or the Sun to the same) is ALSO 8 light minutes away. It should be obvious that the moon is nowhere near that distance away, else its orbit would intersect the sun. (The moon is roughly 1.2 light seconds or 1/4 million miles away)
L3 is listed as being on exactly the opposite side of the sun from us, so that, again, is nowhere near the orbit of the moon. I'll leave the math as a exercise to the reader.