definition of rectangle I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle.

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral.

Because that the sum of the angles of a quadrilateral is 360 degrees is a theorem, it is little ambiguous for me to define a rectangle with a theorem and some calculation (360/4=90).
And if that's is the case, is it fine to say a regular polygon (except equilaterial triangle) is a polygon which has equal sides and equal inner angles?  

The definition is correct.
It is normal to define objects after a theorem, even if at first it seems counterintuitive. In advanced mathematics it happens all the time that a result (like a theorem) permits you to write a definition that otherwise wouldn't make sense.
There is no ambiguity because:


*

*Quadrilaterals are defined before rectangles;

*You can prove that for any quadrilateral, the sum of internal angles is $360°$;

*Therefore, there exist only one type of equiangular quadrilateral: the one with four right angles. No other one!


You can call this unique case "rectangle", with no ambiguity.
(The only ambiguity, strictly speaking, is that there are many rectangles, of different sizes and ratios. But that is not a problem.)
The definition you gave for regular polygons is also correct.
