Look at me as an example:
I studied chemistry for eleven semesters (1984 - 1990) and then physics for four semesters (1990 - 1992), but I did not finish any of those studies. I have no degree in chemistry or physics. But I have a degree in computer science (2006 - 2009, I'm a Bachelor) and now I am a freelancing software engineer who has his own one-man software company.
I can say truly, and without lying:
Ich habe Physik studiert. (I studied physics.)
Ich habe Informatik studiert. (I studied computer science.)
But it would be lying to say:
Ich bin Physiker. (I am a physicist)
This is not my profession. I am not a physicist. This is not what I'm doing to earn my money. And a »studierter Physiker« is technically just a special kind of »Physiker«. It is a physicist who did study physics, but studying is a compelling requirement to become physicist, so a »studierter Physiker« is in fact the same as a »Physiker«.
I even wouldn't dare to say:
Ich bin Informatiker. (I am a computer scientist)
As I already said before: I studied computer science, and at the end of my study I gained the degree "Bachelor of science and engineering". But creating websites and apps and maintaining a web server and a mailserver is not exactly science. I'm not working as a scientist, so I wouldn't say that I am one. And again: a »studierter Informatiker« is nothing else but an »Informatiker«.
But this is not clearly defined. Some other people might call me a »studierter Informatiker« because in their mind this is a synonym for »jemand, der ein Informatikstudium abgeschlossen hat« (someone who has completed a computer science degree). I think, that this point of view is acceptable.
Some even might say, that I am a »studierter Physiker«, because they use this term as a synonym for »jemand, der Physik studiert hat« (someone who has studied physics). I think, that using the term »studierter ...« in this manner is not acceptable. It is missleading.
A word about laws (addendum):
This board here is about the language German. This is an official language in the European Union (EU), in Germany, Austria, Switzerland, Liechtenstein, Belgium, Luxemburg and Südtirol (a part of Italy). So when you talk about laws in this language board, you have to discuss the laws in all of those nations. (This here is not a board about Germany or any other nation).
I'm living in Austria (and therefore of course also in the EU). I have only little knowledge about German laws. Very often they are very similar to Austrian laws, but sometimes they are also completely different.
I just know that academic degrees in Austria are different from the degrees you can gain in Germany. (For example: In Germany there is a degree "Lizenziat" that doesn't exist in Austria). In Austria academic degrees and some job titles are protected by laws. This means, you can be punished by Austrian authorities if you call yourself a »Ziviltechniker«, because »Ziviltechniker« is one of the protected job titles. I think this is very similar in Germany too (although there is no »Ziviltechniker« in Germany).
In Austria neither »Physiker« nor »Informatiker« are protected job titles, so everybody can call himself legally a »Physiker«, and I think this is true for Germany too.