An alternative definition to the verb form of the word "claim" could be: to assert or affirm the truth or existence of something. In this context, it aligns with the idea of making a statement of the truth.
When someone claims something, they are stating that they have a right to it. For example, a person might claim ownership of a piece of property, asserting that it belongs to them.
Claiming can involve making a statement of fact or truth. For instance, if someone says, "I claim that I saw the accident," they are asserting the truth of their statement, essentially affirming that they witnessed the event.
Claiming can also be about expressing confidence in the truth of a particular statement. When someone says, "I claim that this product is the best on the market," they are affirming the truth of their belief in the product's superiority.