You may want to look more closely at the use of the term "person" in the development of Trinitarian doctrine. The formula used in our statement of belief has two components--three persons, one God:
There is one God: Father, Son, and Holy Spirit, a unity of three co-eternal Persons.
The One God describes the Father, Son and Holy Spirit. Together they comprise One God. Person is not always used in the same way as we think in Trinitarian discussion.
The co-eternal aspect is meant to deny the Arian views of James White, Uriah Smith etc.
One of the more famous historical statements clarifying the use of the term person in Trinitarian discussion is the Athanasian creed. Here is a portion of it:
4. Neither confounding the persons nor dividing the substance.
5. For there is one person of the Father, another of the Son, and another of the Holy Spirit.
6. But the Godhead of the Father, of the Son, and of the Holy Spirit is all one, the glory equal, the majesty coeternal.
7. Such as the Father is, such is the Son, and such is the Holy Spirit.
8. The Father uncreated, the Son uncreated, and the Holy Spirit uncreated.
9. The Father incomprehensible, the Son incomprehensible, and the Holy Spirit incomprehensible.
10. The Father eternal, the Son eternal, and the Holy Spirit eternal.
11. And yet they are not three eternals but one eternal.
12. As also there are not three uncreated nor three incomprehensible, but one uncreated and one incomprehensible.
13. So likewise the Father is almighty, the Son almighty, and the Holy Spirit almighty.
14. And yet they are not three almighties, but one almighty.
15. So the Father is God, the Son is God, and the Holy Spirit is God;
16. And yet they are not three Gods, but one God.
Adventists do not go by creeds, but the language used in our fundamental beliefs statement has a history.
Notice Athanasias says there is One God, that the Son is God, Holy Spirit is God, etc. but they do not say the Son is the Holy Spirit, etc. The persons are not confounded, neither is the "essence" (another important Trinitarian term) divided.
The tension often found in Trinitarian discussion is an avoidance of two extremes--modalism and tri-theism.
Tri-theism--there are three separate gods
Modalism-God is just putting on different masks, expressing Himself at various times as the Son, Father and Holy Spirit.
Both of these are rejected in the traditional view.
Related to modalism is the other idea you have expressed, that the Father suffered with the Son on the cross. This is not the traditional view either, but a heresy known as Patripassianism. The traditional view does not hold that the Father suffered on the cross or died on the cross.
However, to avoid Nestorianism the traditional view does hold that the Divinity of Christ died on the cross, or else there could be no atonement, as only God could atone for His law.
Needless to say the subject is complex. But the traditional view holds that there is a One God who is a unity of three co-eternal persons. There are not three gods but One God.