polygon & quadrilateral
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That is not really true. A trapezoid is a particular type of quadrilateral and that, in turn, is a particular type of polygon. That does not make them alternative names for a trapezoid.
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A trapezoid is called a trapezium in the UK (because the UK trapezoid is scalene). In any case, a trapezoid is also a quadrilateral, in that it has four sides.
All the names to classify a trapezoid are a trapezoid and a quadrilateral.
A trapezoid is a quadrilateral shape that has four sides of unequal lengths two of which are parallel to each other. An isosceles trapezoid also has two parallel sides but with two other sides being of equal length.
A trapezoid is a shape which has parallel lines and two other lines connecting the parallel ones.
A trapezoid, a trapezium, an isosceles trapezoid or trapezium and it is a 4 sided quadrilateral.