Not quite.
I agree that copyright exists upon creation (if applicable).
The difference is if it was registered prior to the violation of copyright. In either case, the copyright must be registered prior to bringing a lawsuit to trial. Section 411(a) of the copyright act requires registration for trial except in the case of a suit for the violation of moral rights (section 106).
Section 411 does allow for "pre-registration" (application filed/fees paid) and that without a completed registration (or ,in fact, with a refusal of registration) you can proceed with a suit. But then a large portion of the case will depend upon the determination of copyright ownership/validity.
We tend to believe "we took the photograph we own the copyright" but that may not be the case. In the case of a completed registration, or a rejection, the bias goes towards the copyright office's decision (which can be questioned/argued, but with limited probability). Without a decision (statement) from the copyright office you open yourself up to many questions as to the validity of your claim...
If there are 20 identical images from 20 different photographers, are they all "original?"
If the photo in question was a re-creation of someone else's work (i.e. you recreate the Mona Lisa almost exactly), is it a derivative work?
If the photograph is largely dependent on another copyrighted piece, do you own the copyright? (i.e. statue, billboard, etc etc)
Without a completed registration the bias is not in your favor. Honestly, I would probably expect most of these questions to go in our favor as the standard for "original" is pretty low. But I wouldn't bet on it in any one case. There have been cases which were decided completely contrary to what I would have expected. I can remember one (which was UK based so less relevant) where the image was completely different but included a portion of the same building and a red bus of the same type as were in another photograph. The second was determined to be a derivative work and copyright infringement of the first. There have been US cases as well, but I don't recall specifics.
And then, once you get done with all of that and your wallet is thinner, you can proceed w/ C&D and actual damages.