First of all, you're REALLY reaching with that last image. It is NOT an Uzumaki whirlpool. It's a very common fire symbol. Used in all Japanese works. Even Hashirama had those emblems on his throne thing. It denotes the land of fire. That's all that symbol is.
I never noticed the other ones, though. And they are, indeed, whirlpool's. Which has become a symbol of the Uzumaki clan. How does that make him an Uzumaki? Isn't it possible, nay more likely, it just shows where the clan got the symbol? Showing heritage. We know that the senju and uchiha are descended from the Sage. We know the Uzumaki are cousin to the Senju. That's why the symbols have been passed down, through the ages. It doesn't make the Sage an Uzumaki. It's further proof of the Uzumaki's relation to the sage, and nothing else.