Don't Arabic-speaking Christians also use the word "Allah?"

NOTE: This is a re-post of this blog.  The earlier post, after being edited, suffered formatting problems which I could not, for some reason, correct.

I've been seeing this argument somewhat frequently, and I wanted to address it here:

 

While this is absolutely true, it is somewhat misleading.  The word "Allah" in the Arabic shares a linguistic origin with the Hebrew word "אֱל֫וֹהַּ" (eloah) That is the base word, "אֱלֹהִ֣ים" (elohim) is the plural of that. 

Eloah, and Elohim, are not names. They are the generic terms for "god" or "gods." For example, in Deuteronomy 6:14-15, we read

14 You shall not go after other gods (אֱלֹהִ֣ים), the gods (אֱלֹהִ֣ים) of the peoples who are around you— 15 for the LORD your God (אֱלֹהִ֣ים) in your midst is a jealous God (אֵ֥ל) —lest the anger of the LORD your God (אֱלֹהִ֣ים) be kindled against you, and he destroy you from off the face of the earth. 

The word "gods" in this passage is the Hebrew word "אֱלֹהִ֣ים" (Elohim), except in one instance where it is shortened to simply "אֵ֥ל". It's the generic term for "god." 

So yes, while Arabic-speaking Christians use the word "Allah" to mean "God," so do Arabic speaking Hindus and Arabic speaking pagans who worship Zeus or Thor or Loki.

If the argument, then, is that since both Muslims and Christians use the word "allah" to mean "god," that must mean they worship the same god, then the Muslims also need to accept the Hindus and Pagans as serving the same god they do. 

This is the logical conclusion of this argument. If you're fine declaring the Muslim god to be the same god as the Hindu gods and also the Pagan gods, feel free to continue making it.

CLARIFYING EDIT:

The argument is usually that since Christians also use the word, then Allah is also the God of the Christians. 

The problem with that argument is that Arabic speaking Hindus or Pagans would also use the word Allah. So if the linguistic similarity is the criterion by which we determine which god is which, we must also conclude that Hindus and Pagans worship the gods of both Muhammad and Abraham. 

Since no Muslim is willing to carry the argument that far, it falls flat.