Abhishek has provided very good explanation I would extend the discussion from "based on" to more generic verb-ed modifiers and verb-ing modifiers - noun modifiers.

When you have a verb-ed or verb-ing modifier at the beginning of the clause, the easiest way to check if it is placed and worded correctly is to see if it makes sense with the subject of the clause.

Choice A: Based on accounts of xyz, scholars have painted abc.
Check - The modifier placement and wording is correct if the following sentence makes sense:
"Subject + is/are + modifier"
Scholars are based on accounts of xyz...- This is incorrect. Scholars are not based on accounts of xyz...Thus, modifier error.

Choice E - Using accounts of xyz, scholars have painted...
Check - The modifier placement and wording is correct if the following sentence makes sense:
"Subject + is/are + modifier"
Scholars are using accounts of xyz...- This makes sense. Thus, this modifier is correct.

Lets consider the case now in which both verb-ed and verb-ing modifiers are correct. Look at OG12#36:

Dressed as a man and using the name Robert, Deborah Sampson joined the army...
Check - Deborah was dressed as a man - Makes Sense - No modifier error
Check - Deborah was using the name Robert - Makes sense - No Modifier error

Thus, as you can see, in this sentence both verb-ed modifier and verb-ing modifier are correct per the context of the sentence.

The underlying theme is always the meaning!!!