an html file does not have anything to do with a php handler so the answer is no the php handler will not make any difference.
By Default a Server will load either an index.php, index.html or index.htm file. If you do not want the Server to load the index.html file then tell it not to load the index.html file in the new .htaccess file in the folder for that website that YOU create for the website that you are calling Server Account B:
# DIRECTORY INDEX FORCE INDEX.PHP
# Use index.php as default directory index file
# index.html will be ignored will not load.
DirectoryIndex index.php index.html /index.phpAt this point I cannot explain these basic website principles any simpler.