[seqfan] Squares in a grid

[Joshua Zucker]

I was reading the entries for A002415 and its friends, and I just
can't figure out http://www.research.att.com/~njas/sequences/A024206 .
 What exactly does "distinct up to translation" mean?  Looking at
http://www.research.att.com/~njas/sequences/a2415.gif I think that
a(4) should be 6, not 5, since there's the 1x1, 2x2, 3x3, the diagonal
sqrt(2), and the two mirror-symmetric ones with sqrt(5) as one of the
vectors.  I suppose those are not considered "unique up to
translation" but then I don't see the difference between this sequence
and A108279.  What makes these two sequences different?  I'm kinda
thinking that the sequence differs from A108279 because the square
with side length 5 parallel to the axes is considered different from
the square with side length 5 as the hypotenuse of a 3-4-5 triangle,
but then I don't know how to say what I mean there exactly.

Maybe what I'm thinking is
A002415: squares
A024206: squares, considered different if they are not translations OR
reflections in the coordinate axes (but rotations ARE considered
different?)
A108279: noncongruent squares

Any pointers on figuring this out would be much appreciated.

Even more appreciated would be some pointers about proofs that the
formulas for answering these square-counting problems are as
indicated, and also any nifty bijective proofs (combinatorial
arguments) that the various other things counted by these sequences
really are equal to each other.

Thanks!

--Joshua Zucker