The etymonline entry for assassin is in agreement with the OED entry for it, and in agreement (as far as it goes) with several other online sources (1,2,3,4,5). The last of those links (from thefreedictionary.com) provides some additional information (highlighted in following) that is not mentioned in etymonline or OED:
Word History: Active in Persia and Syria from the 8th to 14th centuries, the original Assassins were members of the Nizaris, a Muslim group who opposed the Abbasid caliphate with threats of sudden assassination by their secret agents. Other populations of the area regarded the Nizaris as unorthodox outcasts, and from this attitude came one of the names for the group [حشاشين (ḥaššāšīn, “hashish users”), or أساسيون (ʾasāsiyyūn)] a word originally meaning "hashish users," which had become a general term of abuse. Reliable sources offer no evidence of hashish use by Nizari agents, but sensationalistic stories of murderous, drug-crazed [ʾasāsiyyūn] or Assassins were widely repeated in Europe. Marco Polo tells a tale of how young Assassins were given a potion and made to yearn for paradise – their reward for dying in action – by being given a life of pleasure. As the legends spread, the word [ʾasāsiyyūn] passed through French or Italian and appeared in English as assassin in the 16th century, already with meanings like "treacherous killer."
If the above is correct, because Marco Polo was Italian, the story he brought back would have spread in Italy first, and from there to France, Spain, England, etc.
A problem with that explanation is that OED shows the Latin form Assisinos in use already in 1237, which predates Marco Polo's birth by around 17 years. The first Crusades began in 1095, around 160 years before Polo.