change sentence: "The search condition of the command (the WHERE clause) is re-evaluated to see if the updated version of the row still matches the search condition."
with sentence: "The search condition of the command (the WHERE clause) is re-evaluated only for updated versions of rows to see if they still match the search condition. Note that this is different from re-executing the command."
I'm indifferent about adding the additional note about this not being a new command start: while the whole of the paragraph makes it clear that once the command begins it waits and then continues the added note isn't going to make things worse.
I do not believe the "only for updated versions of rows" is an improvement - the immediately preceding sentence already says exactly that. I'd rather use fewer words in this already quite long paragraph.
