That is an assumption. The law says nothing about where the "Party" shall be. If it implies anything about the whereabouts of "the party", it implies that he is in the "free state" claiming his slave.
You would make the condition that the slave owner be back in the state under the laws of which his slave is held, but the law does not specify this as a requirement. Therefore it isn't a requirement.
But in accordance with the meaning.
If you are liberally "creative" with the meaning. If you go along with the plain text and in context of 1787 meaning, it doesn't mean at all what you claim.
Your reading comprehension problems are duly noted.
Your desire to deliberately make up interpretations that you prefer is duly noted. Neither the plain words, nor the manner in which the law was applied in 1787 agree with your claims. Furthermore, the slave states would not have agreed to your "interpretation" when the constitution was ratified.
No person held to service or labor in one state...
For the purposes of our example, Kentucky.
...under the laws thereof...
Said laws allowing slavery.
....escaping into another...
Escaping, not being brought. Escaping. Fleeing. Illegally beating feet and leaving Kentucky.
...shall, in consequence of any law or regulation therein...
For the purposes of our example, an Indiana law prohibiting the ownership of slaves by residents of Indiana.
...be discharged from such service or labor, but shall be delivered up on claim of the party to whom such service or labor may be due.
In our example, the owner in Kentucky. Where slavery is legal. As opposed to Indiana. Where slavery is not. Therefore the captured runaway is returned to its owner in Kentucky, not to its owner now living in Indiana, because slave ownership is illegal there.
As further proof of how fallacious your argument is, look at the Confederate Constitution. It found the need to modify their own article IV, section 2, clause 1 to read: "The citizens of each State shall be entitled to all the privileges and immunities of citizens in the several States; and shall have the right of transit and sojourn in any State of this Confederacy, with their slaves and other property; and the right of property in said slaves shall not be thereby impaired. Likewise they saw the need to modify section 2, clause 3. In addition specifically identifying slaves, where the U.S. Constitution had merely said, "escaping into another" the Confederate leaders felt the need to clarify it by saying "escaping or lawfully carried into another." Why did they see the need to make those changes if a clear reading of the Constitution meant that no state could outlaw slavery completely within its borders?