(Corrected spelling mistake)
These different scripts evolved in different parts of the Middle East. In the Aramaic tab on the webpage you sent me, I believe all those three different alphabets are ancient and no longer in use.
However, if you look at the Syriac or Turoyo tab, it says "Syriac", which I believe is the one alphabet to use.
I could be wrong (I'm no linguist, nor do I speak Aramaic). Anyhow, even among the current Aramaic dialects, there are different alphabets, Estrangelo, Mandaean script etc. I'm not entirely sure which one is the most widely used.
Also, I don't think there is any real difference in the spelling, other than perhaps vowel markers.
Also, I'm now doing a Y-DNA 37-marker test, I hope it will prove fruitful. Perhaps I'll do a mtDNA study on my father in the future like you did.