I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals should be partitional or hierarchical. It turns out that until relatively recently (say, 120 years ago) partitional definitions were the norm.
For example, Euclid’s own definitions were unequivocally partitional: in the Elements, Book I, Def. 22 he wrote (translation is Fitzpatrick’s 2007 edition from Heiberg’s Greek text; emphasis added):
And of the quadrilateral figures: a square is that which is
right-angled and equilateral, a rectangle that which is right-angled
but not equilateral, a rhombus that which is equilateral but not
Thus, for Euclid a square was neither a rectangle nor a rhombus. Not surprisingly textbooks that followed Euclid closely tended to adopt this classification up through the middle of the 19th century. For example the 1849 edition of Playfair’s Elements of Geometry defined "square", "rhombus" and "oblong" (the word "rectangle" does not appear at all) as follows (emphasis added):
Of four sided figures, a square is that which has all its sides equal
and all its angles right angles... An oblong is that which has all its
angles right angles, but has not all its sides equal… A rhombus is
that which has all its sides equal, but its angles are not right
Similarly Brewster’s translation (1834 and 1851 editions) of Legendre’s Elements of Geometry defined the quadrilaterals as follows (emphasis added):
Among the quadrilaterals, we distinguish: The square, which has its
sides equal, and its angles right angles. The rectangle, which has
its angles right angles, without having its sides equal. The
rhombus, or lozenge, which has its sides equal, without having its
angles right angles. And lastly, the trapezoid, only two of whose
sides are parallel.
Notice that in this last passage, in addition to defining "rectangle" and "rhombus" in an exclusionary way, the word "trapezoid" is also defined so as to exclude parallelograms.
However, in the second half of the 19th century this tradition of partitional definitions began to give way to the hierarchical classifications currently in use. The 1872 edition of Legendre’s Elements from the same publisher replaced the exclusive definition of rectangle quoted above with an inclusive one, defining a square as an "equilateral rectangle". Despite this, "rhombus" was still defined so as to exclude right-angled figures, and "trapezoid" was defined as "a quadrilateral which has only two of its sides parallel."
From ca. 1850 through 1910 texts were all over the place with respect to the definition and classification of quadrilaterals:
- Hill's (1884) A Geometry for Beginners used exclusive definitions in all cases: "The sides of a parallelogram may be either equal or unequal, and the angles may be either right or oblique ; so there are in all four kinds of parallelograms : the oblique unequal-sided parallelogram or RHOMBOID.. the oblique equal-sided parallelogram or RHOMBUS… the right unequal-sided parallelogram or RECTANGLE… and the right equal-sided parallelogram or Square."
- Hall & Stevens' (1891) A Text-Book of Euclid’s Elements for the Use of Schools defined "rhombus" so as to exclude squares, but "rectangle" so as to include squares. (In other words it followed the same conventions as the 1872 English-language edition of Legendre’s text.)
By 1900 or so most textbooks had settled on inclusive definitions for "rectangle" and "rhombus", so that a square was a species of both.
However even after this the convention persisted (and still persists!) to use an exclusive definition for "trapezoid". For example, Wells & Gart’s (1915) Plane and Solid Geometry explicitly stated that "a rectangle is a special parallelogram" and "The square is a special rectangle and also a special rhombus"; however, it continued to define "A Trapezoid is a quadrilateral which has one and only one pair of parallel sides". Similarly Auerbach's (1920) Plane Geometry had "A trapezoid is a quadrilateral with one, and only one, pair of parallel sides." The SMSG (1961) Geometry defines a trapezoid as "a quadrilateral in which two, and only two, opposite sides are parallel", and the two modern Glencoe-McGraw Hill Geometry textbooks that I have access to follow suit.
So, to make a long story short: the hierarchical definitions that are now commonplace first began to appear in the mid-19th century and became standardized in the late 19th and early 20th centuries, but the transition was a gradual and somewhat messy one, and is still incomplete today, at least with respect to trapezoids.
Having said all of this, it would not surprise me at all if there are places in the world where partitional definitions for rectangle are still in use – indeed in a setting where Euclid's Elements is still taught, I would expect that to be the case.
Squarewas a subclass of
Rectanglebecause it just made sense to teach things that way! $\endgroup$