Your argument relise on the difficulty of stating when one phase of something changes into another.
What you seem to not be able to understand is that the words fetus and baby are nouns that we use to describe the same thing. Just like embryo, toddler, child, blastocyst, adult, zygote, teenager, and old geezer. All are words that we use to describe a human being at one stage of his life or another. A fetus isn't a different sort of creature than a baby. They are both accurately described as a child.
Give me a definition of person that encompasses a few cells with no brain.
Person is a legal term. If you refer to Black's Legal Dictionary. The very dictionary that the justices on the Supreme Court use, you will find that person is defined as "a human being". That's it. All human beings, in the eyes of the law are persons.
I know this is going to be over your head, but I will say it anyway. Person is a matter of kind, and not degree. That is, person is the sort of creature you are rather than a result of the degree to which you manifest your potential. There are all sorts of things you can do to make yourself a better or worse person, but nothing at all that you can do to make yourself more of or less of a person.
If being a person were a matter of degree, that is if personhood were a product of the degree to which you manifest your potential, then anyone who is smarter, stronger, bigger, better looking, etc than you would be more of a person than you. That isn't the case. You are exactly as much a person as anyone else in the world because you are exactly as much a human being as anyone else in the world, and unborns, at any stage of development are exactly as much a human being as you. Immature, but human beings none the less.
The fact that you can't seem to wrap your mind around that biological fact speaks to your own intellectual shortcomings and doesn't alter the facts at all.