Unless things have changed recently, only the very first extension counts as a 90 day report and gets you a slip in your passport with the next reporting date. After that, extension dates and 90 day report dates are generally not connected to each other. (I say "generally" because I have heard of immigration occasionally doing the report when someone comes in for an extension, but that's not the general practice.)
Totally agree, I've been very happy with it - it's made life much less stressful, and being able to use Fast Track makes travel really easy. But I'm still amazed at how many staff still haven't gotten that memo!
I know. What I'm saying is that an officially certified copy like the one in the photo can be "chain authenticated" using the procedure I described in order to be usable for any official purpose. The certified copy from the county could generally be used "as is" almost anywhere in the US (other than some court proceedings), but some authorities can and do insist on the fully authenticated version.
That's talking about a Thai marriage certificate - the OP is asking about a US marriage certificate. The US Embassy can authenticate the signatures of certain MFA officials, because they are given copies of the names and signatures of those officials to keep on file. The embassy has no way to authenticate the signatures of state officials in the US.
They're using the wrong terminology - what they really want is an authenticated copy, not a notarized copy. To fully authenticate the document, you would need to have the seal and signature shown in the photo authenticated by a high level office of the State of Oregon, probably the Secretary of State's office. That officer's signature would be authenticated by the US Dept of State in DC, and THAT signature would then be authenticated by the Thai Embassy in DC. It's a very cumbersome and time consuming process that is total overkill for something as minor as a visa extension, but they can legitimately request it. (There's an international convention that greatly simplifies the authentication process, but Thailand isn't a signatory.)
Other people have said that their embassy notarized their marriage certificate. If the embassy will let you do a self-serving affidavit stating that the certificate is genuine, the MFA in Bangkok could authenticate the signature from the embassy and immigration would probably be happy. As a legal procedure that's totally meaningless, but all they really care about is the form, not the substance.
A notary seal can be raised or just a rubber stamp, depending on the purpose for which it's being used. (Someone who plans to fax or scan a document often prefers a rubber seal.) I was a notary for my entire career and had both types of seals.