I searched Changes to the Box Top Rules, 1949 - 1999 and found the following clarification was made in 1953: 1953: If a word is formed that covers two premium WORD squares, the score is doubled and then re-doubled (4 times letter count), or tripled and then re-tripled (9 times letter count) as the case may be.
Nowhere in the rules was there any mention of being able to use the same premium square twice.
The Diamond Anniversary Edition rules includes some interesting variations, but does not mention your scoring rules.
Personally, I would find your scoring variation confusing and prone to error, since Scrabble tiles are not transparent, and therefore a novice may have trouble realizing that they are reusing a premium word square.
boardgames.stackexchange.com/users/632/ghoppe
I have certainly played Scrabble games with people that thought you could re-use bonus squares, though I haven't heard of the distinction you make between word and letter bonus squares.
I think this is not so much a common variation as it is a common misconception (like Free Parking in Monopoly).
I have had reasonably-good success weaning people of their attachment to this non-rule by purposely making solid blocks of tiles that have to be carefully lifted and replaced for every word I play :-)
boardgames.stackexchange.com/users/1191/Mark Bessey
Your question isn't completely clear.
Once a word has been played.
the bonus can't be used again in a later turn, for whatever reason.
If, however, you play a letter that makes one word across and another word down (usually filling in a block of four), then you score both words; if the letter is on a bonus square, presumably you count the bonus twice.
boardgames.stackexchange.com/users/1644/Tim Lymington
To answer your question directly - yes, I have come across many people, many times, who were unsure or even adamant that bonus squares should be considered following their intial play.
Does that make it alright?
Sure, if every player participating preemptively agrees upon it, much like they must agree to limitations of time, specify what book will settle their disputes should they arise and so forth.
Does that make it wrong?
Technically, yes it is incorrect but who cares?
Anyone considering playing this way is obviously no Joe Edley
so naturally they wouldn't!
The beauty of Scrabble lay in its malleable nature!
In no two parts of the world are there identical rules and conventions set in place that agree with the next game's set of rules; Everybody uses a different dictionary.
Everybody subscribes to their own personal rendition of the game.
I have encountered many people over my lifetime of scrabble who have been confused and or were insistent on this variation you bring to light being The Rule.
Though I do not agree that this is a proper rule of the game as its creators and overseers have seen fit to include - I do not see any just cause for stifling your own enjoyment of the game!!!
I say play it however the hell you please and let no other signal guide than those variantions that move you :*)
boardgames.stackexchange.com/users/8048/TheGenesisBloke
I'm playing right now.
And the rules state it CAN be done.
Rule 7.
When two or more words are formed in the same play, each is scored.
The common letter is counted (with FULL premium value, if any) for EACH word.
boardgames.stackexchange.com/users/26272/Dustin Taylor
Is there a popular Scrabble variation that:    lets you play double/triple word scores twice, once vertically and once horizontally?   does NOT let you play double/triple letter scores twice?    This is how I've always played, so it surprised me to learn that official Scrabble rules do NOT permit using double/triple word scores twice.   Perhaps this was true in an older version of Scrabble (I started playing in the 70s).
Scrabble variation: use double and triple word scores (but not double/triple letter scores) twice?
boardgames.stackexchange.com
<scrabble>
boardgames.stackexchange.com/questions/9118
boardgames.stackexchange.com/users/None/
This is possible in the rules, but a clarification was made in the 1950s. A lot people play this way. Some see this as a common misconception. Yet, the rules state it CAN be done.  When two or more words are formed in the same play, each is scored.  In Changes to the Box Top Rules, 1949 - 1999  the following clarification was made in 1953:  If a word is formed that covers two premium WORD squares, the score is doubled and then re-doubled, or tripled and then re-tripled as the case may be.   
The rules state it CAN be done.  When two or more words are formed in the same play, each is scored.  In Changes to the Box Top Rules, 1949 - 1999  the following clarification was made in 1953:  If a word is formed that covers two premium WORD squares, the score is doubled and then re-doubled, or tripled and then re-tripled as the case may be.    Some see this as a common misconception, but many people do this. 