I think you've definitely capture what the Trinity consists of in your 6 statements, although you need to add a seventh: "There is only one God." The way trinitarians avoid tri-theism and contradiction is by making a categorical distinction between "personhood" and "being."
To illustrate this, there are some beings that are not persons at all. A rock, for example, is a being, but it's not a person. A human, on the other hand, is a being that is also a person.
If you tweak #4, 5, and 6 of your statements to say, "The Father is not the same person as the Son," etc., then you can deduce the Trinity from those 6 statements.
From 1, 2, and 3, combined the the 7th statement I mentioned above, it follows that the Father, Son, and Holy Spirit are the same God. That is to say, they are the same being.
From 4, 5, and 6, it follows that the Father, Son, and Holy Spirit are distinct persons.
Since "being" is categorically distinct from "person," it is possible for one being to be three persons. I mean, if you can have one being (like a rock) that is zero persons, another being (like a human) that is one being and one person, there shouldn't be any problem with there being one being that is three persons. It would be a tri-personal being, i.e. one being that is three persons. It's strange, but it's not contradictory.
To have a contradiction, you'd have to say one person is three persons or that one God is three gods, but that's not what the Trinity is. To have tri-theism, you would have to say there are three beings, i.e. gods, but that is also not what the Trinity says.
If the Trinity said that God is one in the same sense that God is three, then that would be a contradiction. But the Trinity says God is one in the sense of being one being, and God is three in the sense of being three persons, then God is not one and three in the same sense. Remember, the law of non-contradictions says that two propositions that contradict each other cannot both be true at the same time and in the same sense.