Q: "I am curious as to how a patent office determines when a website or public disclosure was first officially made."
R: You should read a few recent court decisions that address how the "on sale bar" is analyzed under U.S. law [visit the links below]. Your own patent attorney will have to evaluate how the applicable rules are applied to your particular situation.
Note: When examining a patent application, the Patent Office does not determine when the claimed invention was first offered for sale. That chore is done only when the validiity of the patent is challenged either in court or in a proceeding before the Patent Trial and Appeal Board.
Q: "... can this public disclousure hinder my changes of foreign patents?"
R: Yes. How has to be determined under the law of the country where you seek patent protection.
The above response 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.
It is good you ask these questions before proceeding. You are smart to find out as this can be the difference in success and failure and it is so true than "an ounce of protection is worth a pound of cure".
A patent office determines public disclosure based on documentary proof. For example, in the USA by regulation all communications with the patent office must be in writing (hard copy or electronic). So you need documentary proof and it needs to be corroborated. The USPTO will accept declarations (a special form of statement) unless there is evidence refuting the statement. Also, the PTO accepts the date of filing as the date of invention absent some other proof, as there is a declaration filed as a required part of the application. That will have the effect of a disclosure date once published or issued as a patent.
As of 2013-03-16 the US is substantially harmonized with other nations, and is as stringent on novelty as any. What you likely refer to is the grace period. In the US a public disclosure made by the applicant less than a year before the filing date is not prior art. In most foreign countries it is prior art. So yes, that public disclosure might very well cost you your chances of foreign patents in many countries. That is why prompt filing is so important in patent matters.
You need to keep in mind that a provisional patent is NEVER published by the patent office, but rather is kept secret and yet confers a right of priority upon which US non-provisional patent applications and foreign patent applications can be based. It is one of the many great advantages of provisional patent applications. The potential for loss of foreign patent protection if you do not understand that this starts a one year period during which foreign patents must be filed, if they are to be filed, is important and indeed crucial if foreign patents are important to you.
This is no business for amateurs. You need a registered patent attorney. No exceptions to that and no excuses will suffice to replace the skills of a registered patent attorney.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Short Answer: Reasonable Inquiry!
The patent office will conduct a search of the prior art based on the resources that the patent office has available to it. In addition, the inventor(s), his attorney, and others with a duty of disclosure are obligated to disclose their knowledge of the prior art to the patent office.
Your point about website disclosure dates is part of the bigger puzzle of how to broaden the search tent to find more of what is out there and relevant to the patent. This also ties into your next question as to foreign novelty requirements; yes, it is possible that the PTO here won't find a reference and a foreign office will, in which case, if the prosecution in the US is still open, the PTO examiner will take that newly found reference into consideration.