Based on the research you have done, as well as the fact that there was no notification that there was a charge for use of the images, I would not pay them. If they continue to contact you, you might retain counsel to send a cease and desist letter.
Please note that I am answering this question as a service through Avvo but not as your attorney and no attorney-client relationship is established by this posting. An attorney-client relationship can only be established through signing a Fee Agreement and paying the necessary advanced fees.
Consult a local IP attorney - most offer free initial consultations. Can locate them here on AVVO under the heading "Find a Lawyer"
This is not intended to be legal advise or as legal representation. I am a California personal injury attorney . Be aware that every state has its own statute of limitations; and statutes & case laws that govern the handling of these matters.
First, don't rationalize. The facts are that you published a photograph for commercial purposes without a license from the copyright owner. That's copyright infringement. Period. You can use the remainder of the facts to persuade a judge that the damages Getty sustained warrant only imposition of a STATUTORY damages award to Getty on the low end of the $750 - $150,000 permissible range. But I wouldn't be persuaded. You chose NOT to use one of the many, low-cost photograph licensing companies and, instead, simply published a photograph you knew you did not have permission to use. That ain't good. Getty makes its money licensing its photographs -- money that it then uses to buy the rights to other photographs. Which goes to photographers who are trying to make a living. You, however, simply took a photograph and used it because you wanted to. Your assertion that you did not know it was a "royalty image" insults reason.
As far as what to do, you should know that Getty only sometimes sues those who infringe its copyrights. You should also know that Getty has a reputation as a bully [see http://www.extortionletterinfo.com ]. Do you need to pay? The legal answer is yes -- you infringed and so the rights holder is entitled to a remedy. The practical answer only you can provide.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
You have already admitted on this web-site that you engaged in copyright infringement. With respect, anyone who is involved in designing web-sites must have a basic education in copyright law. Copyright protection arises automatically for any work that is fixed in a tangible medium of expression. Registration is necessary as a pre-requisite to bringing suit, but it is not necessary for protection. Further, copyright owners are no longer required to affix notices on their works---this is because the law for decades has made it clear to everyone that every work is protected by copyright. There is simply no excuse for lack of awareness of these basic fundamental legal concepts. Your suggestion that "Go Daddy" or other such companies should be notifying its users of these things is, frankly, an attempt to evade your own responsibilities to be familiar with the law applicable to your activities, professional and personal.
Getty Images is a very legitimate business, which generates substantial licensing fees for the photographers who enrich our lives by providing images. When you use an image in this context without paying a license fee, you undermine the confidence of the public in the integrity of our entire legal system. The internet is not the wild west, where you can grab anyone's content and use it for whatever purpose turns you on.
Copyright law provides for statutory damages of no less than $750 up to $150,000 (in the event the infringement is willful) using an infringing "work". Further, copyright law permits the copyright owner to recover its attorneys fees if someone in your situation forces the copyright owner to litigate and if you have no objectively reasonable defenses. You have no objectively reasonable defense to this claim.
But you should not immediately get out your check book. You not only need to settle with Getty, but you need to force Getty to enter into a written settlement agreement that releases you from further liability for your infringing misconduct. You need a lawyer to draft a settlement agreement with Getty that resolves this in a manner that stops them from coming after you----don't pay them until they agree to release you from any further claims for damages or fees. And such documentation including the release language needs to be drafted by legal counsel. You need to retain legal counsel to settle this. The horror stories in the press all relate to situations where people settle with Getty on their own without retaining legal counsel. Your counsel may get Getty to reduce somewhat the settlement amount, but realistically you will have to pay your counsel more than you will be paying Getty. But failure to get legal counsel in this situation is almost always a mistake. There are quite a few lawyers who will handle these negotiations for a reasonable flat fee (I do this on occasion).