I think this calls for a definition of terms. Otherwise, ten people could be talking about a bribe and meaning at least nine different things.
Mr. Merriam and Mr. Webster tell me that a bribe is: a) money or favor given or promised *in order to influence the judgment or conduct of a person in a position of trust*, and b) something that serves to induce or influence. That's a pretty wide-open definition, when it comes to everyday behavior.
But, generally speaking, "I'll do this for you so that you'll do that for me" could be interpreted as a bribe. At least I imagine that a six-year-old would look at it that way. It isn't *necessarily* bribery, but it has that general outline - "I'm buying cookies today, and so you will of course eat your veggies in order to get some."
Keep in mind that a consequence can be neither bribery nor reward. If I have a salaried job and do that job well for two weeks, I receive a certain amount of money. The money isn't (legally) a bribe to make me do the work. Nor is it a reward to praise work appropriately done. It's a legitimate consequence, and it's perfectly fair.
And it's fair, perhaps, if Paul and Penny know that they must finish their veggies if they are going to see the cookies. Cookies have become an agreed consequence of eating all the veggies, and Paul might consider it proper justice if the cookie wasn't forthcoming because he shoved the green beans aside or fed them openly to the dog. (Along that line, to say, "Well, you ate your beans but you didn't fold your napkin, so there's no cookie" would be injustice. So would, "Well, I'll give you a cookie anyway.")
It's also well to remember another term: extortion. It's bribery if you say, "If you act nice to your Aunt Rose this afternoon, you can stay up an hour late tonight." It's extortion if the child uses the routine on you: "Don't expect me to be nice to Aunt Rose unless you let me stay up tonight. Write up an agreement to that effect." That's automatically out of line.
A reward for perseverance, obedience, or some sort of very good action (even taking a nasty medicine) is sometimes fine. It might be best that the reward be more like an "award" - say, a sticker or a ribbon or a mark on the chart - rather than anything tangible. (I might make an exception for taking the nasty medicine.) And a reward usually isn't announced in advance; you look for the good behavior you've been teaching, praise it, and back up the praise with a reward *this time.* (That could include the good behavior of doing the right thing on the potty.)
We all know that bribery can be useful on occasion. "We were just invited to go to the pool with the Jones family. IF your house jobs are done by ten a.m., we can go." That's a sort of bribe, because the mention of the reward in advance is a definite incentive to do the chores quickly. But of course you have to follow through as promised. And, since you all have VERY smart children who want to be lawyers when they grow up, watch for, "Hey, Mama, does this mean that if we have our chores done by ten every day this summer, you'll take us to the pool?" That's edging on a bit of manipulation by the chore-doer!
But bribery works best when done sparingly. The law of diminishing returns fully applies - the more often you do it, the less you get the results you hope for.