As a basic doctrine of the Trinity this is what I hold:
1. God is One.
2. The Father is God
3. The Son and the Father are One.
4.The Spirit is the Spirit of God (God’s Spirit).
I find this explanation sufficient to deal with passages. So now let's see if I can back this up as stated in Scripture or if it is, as you say, merely implied:
"Hear, O Israel! The LORD is our God, the LORD is one! Deuteronomy 6:4
Now a mediator is not for one party only; whereas God is only one. Galatians 3:20
one God and Father of all who is over all and through all and in all. Ephesians 4:6
yet for us there is but one God, the Father, from whom are all things and we exist for Him; and one Lord, Jesus Christ, by whom are all things, and we exist through Him. 1 Corinthians 8:6
I [Jesus] and the Father are One. John 10:30
In the beginning was the Word, and the Word was with God and the Word was God. John 1:1
For in Christ all the fullness of the Deity lives in bodily form Colossians 2:9
Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom Corinthians 3:17
This is what the Lord says – Israel’s King and Redeemer, the Lord Almighty: I am the first and I am the last; apart from me there is no God. Isaiah 44:6
The Spirit of God has made me, and the breath of the Almighty gives me life. Job 33:4
And the Spirit of God was moving over the surface of the waters Genesis 1:2
My spirit shall not strive with man forever Genesis 6:3
You are wrong, David. What I hold as a foundational definition of the Trinity is actually stated in Scripture - not, as you say, implied.
But you cannot do this with Penal Substitution Theory. You can only tell us that Christ died for us which implies God punished Jesus instead of us. Consider how much error has seeped into our faith by this line of reasoning. You cannot argue against any heresy consistently because your opponent can simply fall back on your support and say "it's not stated, but that's what is said...it's implied".
That is the danger when we gravitate towards elevating our own reasoning and understanding as if it were what Scripture actually "says". You end up on sandy ground. You end up elevating yourself to the position of God (you, not Scripture, becomes the "word" of God by explaining what God "really" means). And people see this. When you argue against them they can identify that your view is just as subjective as any other religion.
That's my caution to you. Stop relying on what you feel God wanted to say, what you feel God implied in Scripture, and instead rest on what is actually written. Hold opinions, but do so on non-essential doctrines. Hold your theories, but never build on them. Your house of cards will not stand to Scripture because in truth it is foreign to the actual text of Scripture.