Under no circumstances I am aware of do any of them certify welders. As I said, they survey only, and yes, the end liability falls to the manufacture builder usually. However; there are many instances were that is not good enough.
Loydds of london is an insurance agency, not to be confused with lloyds register, or germanischer lloyds. None of them have direct ties to each other.
GL, BV ABS etc are typically contracted by a governmental agency or Insurance agency such as loydds of london.
IACS companies got their starts in maritime however; they now cover LNG sites, refineries, in some cases nuclear plants, bridges etc. and of course shipping, classifying ships etc. You need to look further into companies.
http://www.gl-group.com/industrial/glo/4830.htm for instance. GL won't reject something, but they won't certify it either and will write a pile of NCR's if needed assuming it doesn't meet the specifications and government regulations.
"I have been the third party" I would call what you describe as third party to be SME verification.
The one thing we agree on is the term "3rd" party is a tricky business, and in my opinion is headed toward centralization because of the mis-use of the term.
Just like AWS, ASNT, and others, It was the people in place at the time of the need that ended up by default being the recognized authority. I foresee a time coming when IACS becomes the defacto standard organization for certification of "3rd parties" because they build on other groups like API, AWS etc. GL for instance does not send out surveyors that are not certified, nor do not have a detailed SME background. Nor does BV or others.
My money is on IACS for those reasons since they've set themselves up a couple of tiers above the other organizations, and since they are the only recognized international organization to perform the function. AWS and others are market specific and therefor have a conflict of interest in that they obviously will favor their own code over say ASME etc. Therein is the need for independence, and the basis for which the IACS companies have set themselves up above the level as it's hard to verify objectively a code you have written. If all people were honorable and skilled, and took pride in their work, SME's 3rd parties, nor even inspectors or examiners would not be needed. Unfortunetly that is not the case, and therefore the rising need, which will drive the effort for an organization to step up and create a centralized third party certification. It's just a matter of time in my opinion.
To say billy bobs test lab is a 3rd party is wrong. At best they are 2nd party.