The word lemon is the original one from Arabic.
Lemons and limes were introduced into Europe by Arabs who planted gardens in Spain and Sicily.
The Arabic form of the word, laymûn, was borrowed into European languages. Arabic got the word from Persian limu, which got it from Hindi nimbu or nibu. When you denasalize the initial n-, it becomes l-. This happened in the word lilac too: Persian got its word lilak (with the diminutive suffix -k) from Hindi nil ‘blue’.
The original source for the Hindi word nimbu or nibu for lemons and limes was one of the indigenous Munda languages of aboriginal India, belonging to the Austroasiatic language family, according to the etymology in the Oxford Hindi Dictionary. Other Austroasiatic languages include Khmer and Vietnamese.
See this page which tracks the etymon for “Citrus acida” (?) in the various Indo-Aryan and Dravidian languages:
http://www.hindunet.org/saraswati/Indian%20Lexicon/citrus.htm
The Munda languages are spoken in India by aboriginal hill tribes, such as Santal, Oraon, Birhor, and Ho. Their relationship to other branches of Austronesian like Mon-Khmer and Vietnamese is a very distant one.
In Italian, too, they have only one word for both lemons and limes. In fact, every language I can think of except English has only one word for both fruits. This goes back to the aboriginal Munda word in India, which used only one name to cover both. The real question here is how English managed to differentiate two words, one for the green one and one for the yellow one.
Oh yeah, botanical taxonomy also differentiates the two: Lemon is Citrus limon and lime is Citrus aurantifolia.
The two fruits not only look different, they taste different too. Seems natural to call them different names. But if you go to South India and check out the lemons, they are small, like limes, and the pulp has a drier consistency more like limes. The limes themselves are less green and more yellowish. In other words, the cultivars are not as differentiated in South Indian varieties.
In modern Persian, although the word limu is used for both fruits, they distinguish limes by calling them limu torsh meaning ‘sour lemon’. Likewise in Modern Arabic, they call limes laymûn hâmiD, ‘sour lemon’.
The American Heritage Dictionary etymology for “lime” says this word came from “Arabic lima, lim, probably from limun, lemon” to explain why we use two different words in English. However, I checked several Arabic dictionaries, including al-Munjid, al-Mawrid, An Advanced Learner’s Arabic-English Dictonary, and Hans Wehr’s Dictionary of Modern Written Arabic. None of them had this alleged word lima, lim, so I am skeptical of the AHD claim. I did find in al-Mawrid by Munir al-Ba’labakki an Arabic translation of “lime” as laym, but this seems to be one of those Lebanese neologisms coined by al-Ba‘labakki after the English word. Maybe because English has differentiated the two fruits, now other languages are trying to find different words for them. English-speaking tourists go to Arab countries and ask how to say “lemon” and “lime,” they’re told the same word. “Really? I can’t believe it, we have two different words for them. Aren’t they different fruits? How come only one word for both?” Etc.