The first name has been shown by one manager as Israel. I have elected the Yisrael spelling upon resolving the data conflicts because it is the more traditional Hebrew transliteration, considering the other Hebrew/traditionally-spelled names in this family. (by Wendy Hoechstetter, 11/30/2012, wah8 at georgetown dot edu)
From my point of view Israel=Yisrael, Moses=Moshe=Moishe=Moische=Moritz, etc.
We researchers must be atune to those differences. The only problem is Geni's search function for which you must spell a profile "exactly" how it was originally written by the person who entered it (unless there was a change made by editing).
Hopefully that will change now that Geni became part of MyHeritage.
I use the name as it appears in documents in the country where the person grew up. I may choose to normalize the transcription to make it easier to search. Leib is how I normalize Lejb. In other words, I try to stick as close as possible to the orthography / pronunciation of the name as it was.
My great grandfather in Suwalki was Eliasz Gersz Frankel (Eliash Hersh in documents of that era. The orthography "sz" is "sh"and I normally keep that spelling. Similarly "g" is the letter used for "h" sound.
Since he moved to Peoria, IL and changed his name to Harry Frankel, that's the name I give him, but I include Eliasz Gersz in the Also Known As and likely also add Hirsh.
I agree that in the AKA field it's a great practice to put Israel, Yisrael to help search.
I *would* use the name as shown on documents of the period, but most of these people are so distantly related to me that frankly, I don't want to expend the energy to research it with so much of the closer-in research I'm interested in as yet incomplete. Therefore, I'm just starting to take my best educated guess in many situations, where there is at least enough evidence to reasonably support that, and to make a note about the discrepancies that the next passengers will see.
I didn't realize that the AKA field was searchable. I've always only used it for things like nicknames and known formal name changes. For spelling variations like this, when I have no idea what the original or what the person actually used were, I've just been noting it in the About section, and now starting to put it in the Discussions in hope that someone else has actual source information.