What is the most common equivalent name for “handal”, a very bitter fruit?

Based purely on the descriptive terms that have now been added to the question, Google Books Ngram Viewer says that the most common term (at least in print) is egusi:

Interestingly, egusi is more common than even terms like bitter apple and wild gourd, which could be used in contexts that have nothing to do with the particular fruit at all.

Also note that vine of Sodom was the most common for a while in the 19th century.

If I add handal into the ngram, it ends up in second-last place, just above desert gourd.

Why is it called vine of Sodom?

Josephus wrote of the fruit of Sodom:

The country of Sodom borders upon it. It was of old a most happy land, both for the fruits it bore and the riches of its cities, although it be now all burnt up. It is related how, for the impiety of its inhabitants, it was burnt by lightning; in consequence of which there are still the remainders of that Divine fire, and the traces [or shadows] of the five cities are still to be seen, as well as the ashes growing in their fruits; which fruits have a color as if they were fit to be eaten, but if you pluck them with your hands, they dissolve into smoke and ashes.

"B. J." iv. 8, § 4

So, this supposed fruit of Sodom looked edible, but was actually inedible. In trying to identify the plant Josephus described, some modern scholars have proposed Citrullus colocynthis. "Turns to smoke and ash when plucked" sounds to me like a pretty poor description of Citrullus colocynthis. It does at least grow in the correct location. And though Josephus doesn't state as much, there's some argument to be made that the plant was a vine.

The somewhat dubious argument is that Josephus's "fruit of Sodom" refers to the same plant as the "vine of Sodom" described in Deuteronomy 32:32:

:כִּֽי־מִגֶּ֤פֶן סְדֹם֙ גַּפְנָ֔ם וּמִשַּׁדְמֹ֖ת עֲמֹרָ֑ה עֲנָבֵ֨מוֹ֙ עִנְּבֵי־ר֔וֹשׁ אַשְׁכְּלֹ֥ת מְרֹרֹ֖ת לָֽמוֹ

For their vine is of the vine of Sodom, and of the fields of Gomorrah; their grapes are grapes of gall, their clusters are bitter;

The idea is that the "vine of Sodom" was a specific plant, rather than a metaphorical kind of vine that you might characterize as being sodomic. That doesn't seem obvious to me. Nor is it obvious that, even if the "vine of Sodom" were a specific plant, the same name would have been used by the writers of Deuteronomy, who were probably writing in the 7th century BCE, and by Josephus who wrote about seven hundred years later.

But fortunately, the question was why do some people call this plant "vine of Sodom" and not whether doing so makes sense.