How can we tell when Scripture is talking about one of the persons of the Trinity and when it's talking about all three?

There seems to be two ways to identify the Trinity in the Old Testament. By the names used to refer to God or by the special quality of the operations that God seems to be doing in the context. Although in general some names seem to refer to the different persons of the trinity, it does not seem, from what I have read on the subject that one can strictly say, ‘Wherever this name is used it means the Holy Spirit, or wherever this name is used it always means the Son.’ In stead each name is a proper name of the’ One true God’. When the ‘name’ is placed within a context indicating the peculiar operations of one person in the Trinity, than it can be said to refer directly to that person. Even if the name generally refers to the Father, under a given context is may actually refer to the Son:

For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace. (NIV Isaiah 9:6)

I will not speak directly about each name, to try and assign which ones are more frequently used for which person, or if they are at all, as that would be a thesis statement for a doctorate in theology. I will only mention that, in addition to the title of the Messiah which we understand points to Christ, the one name of God that most commonly is assigned to Christ is 'Jehovah'. But as in the quote above and other references, the following 'name samples' appear to be used for Jesus Christ specifically on occasion: Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace, Adonai, (Psalm 110:1),Elohim, (Psalm 45:6,Hebrews 1:8)

The main subject to consider is not the specific names but the operations of God and how each person of the Trinity has distinct operation in the relation to the other persons. When the Old Testament refers to ‘God’ in general, then no particular person of the trinity is being identified because all three persons of the Trinity are undivided in their operations and roles. The Father, Son and Holy Spirit as acting by the same exact and undivided will, wisdom, power. All three are the author of the operation. God is generally referred to in this way because of the perfect Unity of all three persons with same essence, likeness and divine nature. However, even when there is no distinct mention of a given person in the Trinity by name, because each person has a distinction from the other two and therefore a relation to them, each can be said to distinctively operate where ever God is referred to as operating. This existence, and mutual relation between each person within the single existence of God has been called their ‘subsistence’. Each person has an eminent distinction and relation to the others in their subsistence. When any special impression is made of the special property of any person on any work; then the scripture is drawing attention to that work as assigned peculiarly to that person. Not that any person is passing the ‘baton’ as it were to the other and no longer acting, because all three are always joined in One performing it, but outwardly and as described in scripture special attention is given to one person of the Trinity at special instances.

As we collect the many places in the Bible where particular persons are mentioned under the acts that are being performed, or specifically and directly by their name, as is often the case in the New Testament, we can recognize patterns, or characteristic properties, to look out for. For example I gleaned these characteristics from John Owen's Discourse on the Holy Spirit, in the third volume of His works, which I read around ten years ago and it still impacts me.

Father: The fountain of the Godhead, beginning of divine operations, power and authority, sending

Son: Eternally born of the Father, proceeding from the Father, condescension, grace and wisdom of the father, subsisting, establishing, upholding, making a consistency, making a permanency.

Spirit: Proceeding from the Father and the Son, condescension, immediate works, concluding, finishing, immediately enabling, completing and perfecting acts and mysterious concluding works.

Note: I have not tried to collect Bible versus that illustrates these characteristics because anyone familiar with the Bible will recognize them. However, just referring to major events like the Father 'sending' the Son to be 'incarnate' and the Spirit 'enabling' the Son to 'establish' the work of atonement and 'send' the Spirit to 'perfect' Christ’s holiness in us, gives a big window into the truth of the matter.

Basically, when the bible talks about God, it means the Trinity or Godhead, while when it says one of their names, such as the Father, it means the Father. But as the Trinity is known--Three in person, One in essence--Then if you talk about the Father, you must be talking in a round-about way, of the Son, and the Holy Spirit, though all are separate.

One link I found, says that there are 4 rules to the Trinity:

1. There are three persons who have a complete divinity, not a part of divinity, therefore, The Father is a complete God, the Son is a complete God and Holy Spirit is also a Complete God.

2. Each person is separated from the others, i.e. the Father is not the Son, the Father is not the Holy Spirit, and the Son is neither of them.

3. There is only one true God,

4. There is an equality characteristic between the persons of trinity

As the Orthodox Church believes:

Following the Holy Scriptures and the Church Fathers, the Church believes that the Trinity is three divine persons (hypostases) who share one essence (ousia). It is paradoxical to believe thus, but that is how God has revealed himself. All three persons are consubstantial with each other, that is, they are of one essence (h*m*ousios) and coeternal. There never was a time when any of the persons of the Trinity did not exist. God is beyond and before time and yet acts within time, moving and speaking within history.

The source and unity of the Holy Trinity is the Father, from whom the Son is begotten and also from whom the Spirit proceeds. Thus, the Father is both the ground of unity of the Trinity and also of distinction. To try to comprehend unbegottenness (Father), begottenness (Son), or procession (Holy Spirit) leads to insanity, says the holy Gregory the Theologian, and so the Church approaches God in divine mystery, approaching God apophatically, being content to encounter God personally and yet realize the inadequacy of the human mind to comprehend Him.

So don't get exasperated when you can't comprehend the Godhead. God are impossible to fully realize and know. Here is a nice story that shows this (Actually only the first part is the story, but it is all good).

In response to this:

My question is, what about all the other verses (especially in the Old Testament) that are not so obviously talking about The Godhead or one of the Persons?

If it just says God, then take them all, they're all One though Three. If it just says the Father, then take them all, they're all Three though One.

By the way, have you ever read or watched Flatland? Somewhat on topic.