Yes, it exactly matches your description that I quoted: "its not required to get into earth orbit, its part of the payload adaptor." The remaining differences between the HAPS and the Electron kick stage don't make the latter any less a stage, rather the reverse.
This does not meet your claim of "precisely like"
"precisely like" means exactly the same in every respect
The Satellite Situation Report tries to get its object identification and nationality information from the owners of the objects, so it's likely the identification of both objects as stages (r/b in SSR terms) comes from Rocket Lab.
All very interesting and all highly illuminating, but ultimately irrelevant because none of this overturns the internationally accepted fact that this was a NZ launch, making this country the 11th country to put a satellite into orbit. This is the hurdle you have to get over. I have the established and accepted facts in my corner; you own the burden of proof to prove your claim.
And the document adds the NZ Government to the organisations that identify it as a US launch vehicle.
Yes, a US owned launch vehicle...
Conceived in New Zealand by the New Zealander who started Rocket Lab in New Zealand
Designed in New Zealand by New Zealand aerospace engineers
Built in New Zealand by New Zealand engineers
Supported by the New Zealand government
Launched from New Zealand at a site owned by a New Zealand/American Aerospace company.
Claimed by Rocket Lab (you know, the actual company that launched it) to be a New Zealand launch.
Until you come up with some verifiable statements, attributable to official aerospace organisations, that New Zealand is NOT the 11th country in the world to launch a satellite into orbit, your claims will not stand up, and I will not be convinced.
NOTE: No registry entries or inferred claims. Only an actual challenge to Rocket Lab's claim are acceptable. Currently, you are the only person on the planet actively challenging this claim.
ETA:
https://www.reuters.com/article/us-lockheed-rocketlab/lockheed-invests-in-rocket-labs-u-s-unit-to-keep-pace-with-innovation-idUSKBN0LY21Y20150302"Lockheed invests in Rocket Lab's U.S. unit to keep pace with innovation"
WASHINGTON (Reuters) - Lockheed Martin Corp said on Monday it had made a strategic investment in the U.S. unit of New Zealand’s Rocket Lab, which is building a carbon-composite rocket, the Electron, to launch small satellites into orbit for less than $5 million.
Ned Allen, Lockheed’s chief scientist, said Lockheed’s investment in the New Zealand startup was aimed at helping Lockheed keep pace with innovation across the industry.
I have quite a number of similar pages. Would you like me to link them?