Elementary Geometry Nomenclature: why so bad? A long-ish wall of text, and I apologize.
Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A student piped up and said, "that's a rhombus." The professor stopped mid-stride, looked at him squarely, and said, "rhombus? That's a stupid word. What's a rhombus? I don't even think that's a word. The word I was thinking of was 'parallelogram'." This was shocking, because this was an American professor, at an American university, and in my American public education, I was taught what a rhombus was in the second or third grade.
Recently, however, I was thinking that maybe my professor wasn't wrong. Consider the naming system for quadrilaterals. The term "quadrilateral" makes some sense: "quad" from Latin for "four", and "lateral" meaning side. And then you get parallelogram, with "parallel" meaning "parallel" and "gram" from Greek meaning "drawn". But then a rectangle is a special case of a parallelogram where the angles are all right angles, which follows clearly enough, and a square is a special case of a rectangle, and important enough to merit its own term.
But then a quadrilateral with only two parallel sides is a trapezoid, which derives from Greek for "table shaped". And then a rhombus is the complement to the square in the special cases of parallelograms -- its angles are anything but right angles!
Confusing yet? We've got the following suffixes describing shapes: -lateral, -gram, -zoid.
We also have triangles, which makes sense because it's "three angles." Yet a "quadrangle" is a region in a university campus.
Increasing the number of sides in the shape, we go from "quadrilaterals" to "pentagons". Ok, now we've gone from the Latin prefix for "four" and a suffix meaning "side" to the Greek for "five" and a totally different suffix. Sometimes we describe the word using a root that means "drawn", and sometimes we describe it by the way that it looks.
And still "rhombus" fits in nowhere in this crazy, convoluted scheme!
To bring this all back to mathematics, and to ask my original question:
Individually, I can find the etymology of each of these terms. But why did the mathematics community adhere to these terms, particularly in elementary education? Did these terms get translated haphazardly from Elements? Is this one of those consequences of the somewhat insular nature of the mathematical community during the Renaissance era? The mathematics community has evolved to be fairly precise in its use of terminology. Why is the terminology surrounding elementary geometry so fragmented?
 A: We also have, for example, add/sum/negation vs. multiply/product/reciprocal. As with natural language (be/is/was, go/went, speak/spoke), the oldest terms tend to be the most irregular, because they became established before the currently used structure emerged.
A: I can't resist adding this, since it's been on my mind in recent months.
This vignette in the OP highlights part of a big problem with modern mathematics education: enshrinement of ideas and terminology as being written in stone. Another closely related thing is the unwillingness of textbook writers from departing from what has already been written. They apparently cannot dream of departing from the herd into more sensible pastures.
I acknowledge, of course, that it's necessary to select a vocabulary so that students can discuss things with you. For the sake of uniformity, teachers tend to stick with what they were taught, both because it is familiar and because other texts do it.
What this has come to in practice, though, is students being rigidly taught that "this is what you call it" along with the undertone that nothing else would be considered correct. As a result we have the patchwork of terminology to forcefeed children with.
I don't really think that this is a result of rigidity so much as it is an ignorance of what mathematics is about. The perception that mathematics is fixed or rigid causes teachers to treat it as such, whereas more mathematically experienced people recognize it is more like a canvas.
One annoying legacy of this insistance on sticking with tradition (that someone else mentioned in the comments) that (in the US anyway) we are stuck with thousands of primary school textbooks insisting that a trapezoid have exactly one pair of parallel sides. It has amazed me that there can be supporters of this position so entrenched, when their viewpoint flies in the face of the rest of the classification of quadrilaterals. (Another thing like this is insisting that kites have exactly two side-lengths, so that squares cannot be kites. This is more rare than the trapezoid thing, but it equally annoys me.)
This usually persists into the minds of secondary school teachers, and hence into their students' minds. Then, a student reaching college might discover that nobody in post-secondary education would entertain such a bad system, and they quickly have to relearn quadrilaterals according to the inclusive system. Or, more usually, the changes are just swept under the rug as stuff primary and secondary schools taught less than ideally.
Anyhow, it is difficult to see a good solution. Authors of textbooks are simply unaware of or unwilling to risk the transition to a 'new' system. Primary and secondary teachers apparently cannot (or will not) be persuaded en-masse that the 'new' system is more coherent and worth adopting.
But that's OK: it's not the worst problem education faces. Maybe overcoming these imperfections are important for developing mathematical minds.
A: We need to get better at naming things and it should follow an understandable pattern for easy comprehension. When we're inconsistent with our nomenclature, it takes the fun out of learning.
Unfortunately, this problem goes all the way back to the numbers we use. Eighty-one is 8 tens and one, which makes sense, but explain "eleven" and "twelve". Then try explaining "thirteen" which is 3 ones and ten (which is backwards).
Where English has unique names for 11 and 12, Spanish has unique names for 11-15, then 16 translates to "ten and six".
Adding these poorly named numbers makes you do an extra step in your head. "Eleven plus seventeen plus eighty-one"? First you have to think about what these words mean, then rearrange their orders in your head by tens and ones, then add them.
Science makes you memorize the names of long dead folks (Ohms, Plancks, Coulombs,...) and computer programming is awful as well. (Did you know that in Unix, the command used to OPEN a file is "LESS"?)
Every subject has things which need new names.
