Thank you very much for your comments! I agree that Geni could have an even more sophisticated distinction between the semantics of "Language/Dialect" versus "Script". However the functionality that I see now is already quite good, and the issue I'm complaining about.
I can see that (as of this writing) a profile can have multiple tabs for different languages. Each language tab already has its own "Title", "Name", "Display name", and "Also known as".
I enthusiastically agree that it would also be better if Geni allowed more details in the name field, particularly allowing to specify the time periods when a particular version of the name was used. However it's a different enhancement, "orthoginal" to the language one.
For example, a profile can have "Russian", 'Ukrainian". 'Belarusan", etc tabs if desired, and each tab already contains all the name fields. Each of these languages uses a version of Cyrillic script, but has a few letters not found in the other languages version of Cyrillic (e.g. Ukrainian ґ, ї, Belarusan ў) . I'm not aware of a context where Geni emcounters a Cyrillic letter and automatically assumes that the text is in the Russian language - if it does, then it's another separate issue to be addressed.
Likewise, it's sometimes very reasonable to have separate tabs for "English" and "German" versions of the name. I definitely don't think it would have been better if the tabs were tagged with a scipt ( "Cyrillic", "Latin", etc) rather than with a language.
I'll illustrate the concrete practical issue that I am trying to address.
What %age of Geni users can read this name: Kazimir Ivanovich Kukharenok or this name R' Nathan Thumim ? (Assuming these examples don't get edited.) A very small one.
I think Geni should encourage profile maintainers to transliterate all the fields in such names (not limited the display name; although titles like "Rabbi" and suffices like "Мл"-> "jr" should get translated) by flagging the lack of a transliteration as an "inconsistency". Geni could further help by programmatically providing a "first draft" of the transliteration, that may need editing/cleanup.
(Please note that automatic transliteration is sometimes not good. A solution where Geni would transliterate names automatically without a human editor confirming and saving the result would not be as good.)