The "whip/chop overflow trick" (or whatever you call it) IS an exploit.
Let's examine why you feel this way.
For example: Protective players should build walls simply because they want more defense (and later castles)... not for a large chunk of gold to fuel your over-expansion, hasten your early tech rate, etc.
This is using an opinion as a means to support another opinion

.
It may be clever (as most exploits are)... but you're playing in an UNINTENDED and unorthodox manner (more so than most other tactics) just to squeeze a little bit more of an advantage out of something. It's cheesy, and if you don't believe so, you're just kidding yourself.
It's not cheesy or any more exploitative than standard play, and if you don't believe so, you're just kidding yourself (see, I can do this too! Isn't it constructive

?). Whether it was intended or not is certainly debatable, given that they added the ability to do this deliberately after the release of civ. Also, we've already established that it's not necessarily an edge at all, but rather has the potential to hurt if used in the wrong situation.
You HONESTLY think the developers intended on players timing mulitple tree chops and whipping population away at the very end of producing a 1/2 cost building in order to create a dramatic overflow of gold?
They allowed it to occur, and have known about it for a long, long time without correcting it. They probably intended it at least as much as the favorite civic bribe bug (as in, since when did intent matter for game rules or definition of exploit?). There are so many things that can abuse the AI that singling overflow out seems a little arbitrary.
It was implemented in case you happened to overflow a building by too much, and to lessen the penalty in case you accidentally did so. It was also probably implemented to LESSEN the micromanagement used when trying to AVOID overflow and loss of resources.
You state this as if it's fact. It isn't, or if it is you've done a sorely poor job in proving it. It is quite difficult to overflow a building too much by accident in this game. If it weren't, this discussion would be moot, because we'd be seeing overflow out of the AI, too.
The game design for Civ 4 usually steers in the direction towards less "tedious" micro. To "whip/chop overflow gold" is micro-intensive and I don't think it's a tactic intended to be used by competitive players.
Micro makes or breaks at the margins. This isn't any more micro intensive than switching a food tile between cities (in fact, it's less since you don't have to enter city screens multiple times). Are you saying tactics like sharing food resources weren't intended also? I don't understand this particular argument.
"Hey, we're not breaking any of the rules or mechanics of the game!"
No... but you're "exploiting" them.
The definition of exploit encompasses virtually every good decision made in civ. Where are you going to draw the line? You can put a pretend line anywhere you want.
As for the argument saying, "Warring, Trading, and simply playing against the AI is an exploit." Why then would you go out of your way and exploit EVEN FURTHER in a very tedious, unorthodox manner?
How tedious is something that takes 10 seconds to execute, TOPS? Are we playing the same game? Warring and trading take more time than this! Do you "HONESTLY"

p) think hitting alt-c a couple times qualifies as tedious micro?!
I'm sure there won't be any official patches coming out again since the developers probably moved on to bigger, better things (Civ 5?!). But if there was another patch, they would surely nerf this and correct it.
Surely, after ignoring it for over a year and multiple patches, they'd suddenly do a 180 and nerf it. Or maybe it's more plausible to assert they wouldn't, based on past experience!
True, but if you ask me, that's exactly what makes it relevant. In a multiplayer game, exploits are fixed so that gameplay doesn't have to depend on using them. It's not about making the game fair, since everyone can exploit; it's about making the game better.
You're missing the point. MP is a realm where you might be getting choked by dog soldiers or even guerrilla archers while trying to whipchop your precious gold. Or maybe they settled that hill city while your chops went into a barracks or granary instead of another settler and some archers. Or maybe they'll convert all your precious gold into their gold as they rape you with chariots. The point is that it's one possibility among many in MP, and improperly USING it can result in a pretty humiliating outcome. In other words, it's pretty balanced.
So again, the question becomes:
How is something that does not materially affect game balance, that has less of an impact than variations in standard play, and that falls within game rules qualifying as an exploit? Or, if you take the broader definition of exploit, how is this different from any number of other effective tactics in civ to which the AI can't appropriately react? Drawing pretend lines in the sand? If you HONESTLY
p) think there's a difference, you're just kidding yourself! Haha!