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A polygon is any plane figure bounded by straight lines. So a triangle is a 3-sided polygon, a square is a 4-sided polygon, a pentagon a 5-sided polygon....and so on.
Yes, a square is an equilateral polygon
A regular polygon
it seems to me that the only polygon of your description is a square
Basically, the meaning of a polygon is closed because the sides are straight. A meaning of a polygon for you: a square. A square has 4 sides that are all straight, a square is a polygon.
In general, a diamond shape is neither a square nor a regular polygon. Only in an extreme form is it a square and therefore a regular polygon.