Is it a rule to put authorship credit on every website we make?

No, there isn't, as far as I know.

You should have negotiated that BEFORE signing the contract (taking the job).

There's no rule but one rule. Do what your client wants. They're the ones paying you.

If it's that important to you to have author info on their site, here's what I suggest: Next time you are negotiating a contract for work, give them two prices. The lower of the two prices (your normal price) requires that they give you credit. The higher price is the one where they don't have to put your author info. You're either getting author credit, which I assume you want because it can lead to more work, or you can get paid a little extra. I've always found when negotiating, it's better to give them a decision to make between two or more variables. The purpose is to remove the yes/no decision from their mind, and replace it with a "yes, but which offer is better" decision. Silly little NLP tactic, but in my experience it tends to work.

You could always list your info at the top of the css stylesheet. At least then, people "in the know" could see who designed the site.

I have to say that work for hire is not the place for advertisements, we do not allow this, now if the contractor wants backlinks from our websites, we can do that, but incorporating a made by link in a work for hire event its just not really what you would expect of a professional.

I have seen this a few times with some contracts I had with a company, they later asked why I never again hired them, (I figured they should have figured it out by now)

Sometimes I am working for someone else, that does not have the time to create products, so naturally having someone else on the bill is not something I would want or need.

Still I can see both sides here having worked both sides of the isle here.

Over all, I think its an obsolete practice.

Thank you all for the suggestions and advise to enable me now think differently. I have already communicated with the client and gave my opinion so they should be OK now.

I can see both sides of the argument here, but and I say this with a lot of alacrity, I have worked with very large corporations and one of the things the unwritten rules of the road is that you will keep your mouth shut about what you did and for how much you did it, otherwise your not going to get more work.

You have to realize that out there in those very tall buildings are people who are stressed out to the max, under those circumstances it is difficult if not impossible to actually be creative under that kind of difficult stress, so they often reach out to someone perhaps like you who can deliver consistently what they want and need.

As you might imagine when a persons job is on the line, they are depending on you to be professional and that means not revealing that you worked on a particular project, usually that means for at least three to five years, but that is usually covered in the contract.

Its the way they work usually you have to have been inside a corporate structure to understand the right way to talk to these people.

But with the right methods you can get good paying work on a regular basis, just dont ever mention the idea that you might publish that you did it.

And this is not just for the OP but for others who may be reading this too, if your into creative services and you want to break into more lucrative gigs, never mention promotional vending.