English and Arabic version at same time

I usually use the ME edition of InDesign to do work in, well, in any language. There’s no real reason I can think of to uninstall and reinstall. The default settings for a new document in the ME edition are set up for starting new documents in a right-to-left language, sure, but you can change all of those with no documents open and all new documents should inherit all of those settings. Happy to go into details about how I’ve set that up, if you are interested.

There certainly are many ways to run more than one language installation of InDesign, but I usually see them in use where you’d actually need to have multiple versions installed - for example, if your primary languages were Japanese and Hebrew. Solutions in these cases are usually operating-system dependent. If you’re using a Mac, you can make a second user and install your second preferred version in the second user and use Fast User Switching. The fact that you said that you tried a separate user profile already makes me wonder if maybe you’re a Windows user? (I don’t know what InDesign does during an installation that would affect multiple user accounts but that is certainly something that happens with Windows installers.)

Mac users can also edit plists to change locale, and Windows users can edit the registry  - if this somewhat hacky solution appeals to you, here’s a Reddit thread describing how to do so to switch locale to Chinese. I’m sure that it wouldn’t be hard to find similar values for English and Arabic locales. 

But solution that I prefer for someone in your situation, assuming you’re a Windows user, is this one - you can simply make a copy of the program folder and give it the correct locale ID. - like en_US for English (US), or ar_SA for Arabic (Saudi Arabia), or he_IL for Hebrew (Israel).