In administator go to "extensions -> languages -> Overrides".

In the filter select the language and instance "site", ie. "English (en-GB) - site".

Click New to create an override.

In "Language Constant" type your "ALT_MY_TITLE_OF_CHOICE" exact the way you did in your module title.

In 'Text" type you translation; "My title of choice".

Save and new (for each next language)

In the filter select the next language and instance, ie. "Nederlands (nl-NL) - Site".

In "Language Constant" type your "ALT_MY_TITLE_OF_CHOICE" exact the way you did in your module title.

In 'Text" type you translation; "Door mij gekozen titel".

Save and new (for each next language)... for as much languages you installed.

Now this module shows for all languages with a translated title.