Proof by assumption

The cardinal virtue in reasoning is a commitment to thinking under the facts and interpreting them logically, without bias or prejudice. 

The first and most straightforward violation of reasoning falls under the fallacy of subjectivism. A person commits this fallacy whenever he claims something is true merely because he believes or wants it to be true. Thus, if p is the proposition in question, subjectivism has the form:

I believe/want p to be true. Therefore, p is true.

Restated: the proponent uses the mere fact he holds a belief or desire as evidence for a proposition’s truth. We can identify the error in this argument by stating the implicit premise. Subjectivism implicitly assumes we are infallible. While the proponent may be, the rest of us aren’t. The thoughts and feelings passing through our minds may or may not correspond to reality. That’s precisely why we need a method of discovering whether they are true.

Caution: the fact someone prefaces a statement with the words “I think” or “I feel” does not necessarily imply subjectivism because this is often a conversational manner of speaking. He may offer an entirely valid argument. 

Another caution: not all statements about thoughts and feelings are subjective. Suppose I am trying to determine whether the emotion I am feeling is resentment or justifiable anger. My thought process may be either objective or subjective: objective if I am open to the evidence, subjective if I decide that my anger is justifiable merely because I can’t bear to think of myself as resentful. 

Restated: subjectivism is not about a statement or conclusion. It’s about the evidence one uses to support the stated conclusion. A person commits the fallacy is committed only when he uses the mere fact that he believes or feels something as a reason for believing it to be true.

Rarely do I hear someone commit the fallacy of subjectivism in the pure form diagramed above or as stated in the question presented. As with most fallacies, the pure, textbook cases are too patently  fallacious for anyone to fall for them. In real life, the fallacies are more subtle. Two other common examples are: “I was brought up to believe in X,” or “That may be true for you, but it isn’t, true for me.”

"It can be expressed like this: "Because I'm correct, my proof is correct"

That can also be considered circular reasoning. 

Logical Form:

X is true because of Y.

Y is true because of X.

Or begging the question

Logical Form:

Claim X assumes X is true.

Therefore, claim X is true.

You are correct, this is a common fallacy.  So common, in fact, it is hardly new or desiring of another name as it fits into a wide array of previously identified formal and informal fallacies.  

In this particular instance what you are calling Proof by Assumption is also known as Alleged Certainty. As Dr. Bo writes:

Alleged Certainty
(also known as: assuming the conclusion)

Description: Asserting a conclusion without evidence or premises, through a statement that makes the conclusion appear certain when, in fact, it is not.

Logical Form:

Everybody knows that X is true.

Therefore, X is true.

Or, in this case, 

I assume (know) X is true.

Therefore, X is true.