Thank you for sharing these details.
Reviewing the current taxonomy setup and the 2 example pages, here are my observations.
Taxonomies like "Arabic Name", "Botanical Name", "English Name", "Plant Family", and "Hebrew Name" etc are independent entities and it makes sense to manage them through their own taxonomy.
However, taxonomies "Phonetic Arabic Name" and "Phonetic Hebrew Name" do not represent independent entities, but are an extension or different names for the items already available in "Arabic Name" and "Hebrew Name". And especially, since it is not possible to create a link between the terms defined in the two taxonomies, it would make sense to store the "Phonetic Arabic Names" and "Phonetic Hebrew Names" within the respective "Arabic Name" and "Hebrew Name" terms, in form of term custom fields.
For example, the attached screenshot "term-field-settings.png" shows how I've added 3 terms fields with the taxonomy "Arabic Names" on my test website.
a). Phonetic Arabic Name
This field will hold the Phonetic Arabic Name for the current Arabic Name taxonomy term
b). Arabic Name Start Letter
This field will hold the starting letter from the name of the current Arabic Name taxonomy term
c). Phonetic Arabic Name Start Letter
This field will hold the starting letter from the Phonetic Arabic Name of the current Arabic Name taxonomy term
The screenshot "example-term-field-record.png", shows a test term entry, with field values.
Note: The fields "Arabic Name Start Letter" and "Phonetic Arabic Name Start Letter", will help you in creating lists with the starting letter filtering, similar to the one on the page "/plant-names-from-florapal/".
If you'll go with this approach, you can remove the custom taxonomy "Phonetic Arabic Name" and fill in the field values in all the terms in the "Arabic Name". And as the "Arabic Name" terms will be already attached to the "Plants" posts, you won't have to assign the "Phonetic Arabic Names" to individual "Plants" posts.
Inside the single plant post template, you can show the information from the attached "Arabic Name" terms using a taxonomy view.
I hope this makes it more clear.