As you already noted, the designer (or their company) holds the copyright for their work by default (not only in Germany but in any jurisdiction adhering to international copyright conventions). If you want to do anything with the graphic, you need to acquire the corresponding rights in some way, usually via the contract by which you hire the designer.
You can theoretically have a contract that doesn’t specify how you can use the work, in which case it defaults to some baseline rights¹, which depend on the jurisdiction. However, I wouldn’t even bother with this case, as it gets easily legally muddy and the baseline rights very likely won’t suffice for you. Neither you nor the designer would want this.
Instead the contract should specify what you can do with the work and what that is up to you to negotiate with the designer – so there is no general answer. The more you want to do be able to do, the more it will cost. You already considered publishing a paper, but what may not be so obvious is:
You may want to publish open access. Most journals use a Creative Commons license. This may be negotiable in some cases, but it’s probably easier to acquire the rights for this upfront.
You may want to use the graphic in outreach or on your webpage.
Journals might use the graphic in contexts where acknowledging the designer may be difficult, e.g., in a graphical abstract.
can I act as if the plot is 'mine' and I 'produced' it? (I don't want to downplay the importance of the graphic designer, but I also don't want to cite/ask him every time I use the image.)
You do not have to academically cite the designer, but acknowledging them is good academic practice. Putting a small line under the graphics in presentations should not cause a distraction, and in papers you can put a few words in the acknowledgments. Moreover, the right not to name the designer may cost extra.
It should go without saying that in no case it would be a good idea to explicitly claim that the work is yours.
A final consideration: I would consider certain digital visualisation skills part of the standard toolbox of scientific communication. I don’t know how challenging your particular problem is, but consider investing the time in learning how to solve it yourself to benefit from the skill all your life.
¹ These are based on the assumption that you wouldn’t want to hire a designer without using the work at all, so there are some implied usage rights.