No, this is not right. You're thinking of "God" as being the name of the shared essence and as being synonymous with "Trinity". You're thinking of God the Trinity as a monad, "God in general", and then God the Father, God the Son, and God the Holy Spirit as a triad, "God in particular". But that is simply some kind of semi-Modalism.
As Trinitarians, we believe that there are 3 types of qualities in God:
- The "essence" is the nature of God, shared by the Father, the Son, and the Holy Spirit; it is the set of attributes that defines the nature of "God". The three persons of the Trinity are all "God" like how every person posting in this thread is "human". We share the same nature, the same "set of attributes" that defines "human". But when God is concerned, His essence is absolutely beyond reach, because His essence is not simply an essence among others to be studied or theorized about, but it is super-essential, it is beyond itself, it is the absolute greatest thing imaginable (or rather not imaginable). In that sense, God is infinitely distant from us.
- The "energies" are the attributes of God, and His acts upon the creation, so to speak. The essence is supremely far, but God manifests Himself in His energies (or even "energy" as He is simple). God's energies are the "potential" of His essence, acting upon the world; but because He is God, His energies truly and fully convey His nature, so that it can truly be said about God that He pervades the world and sustains it with His own being, without falling into pantheism (the doctrine that the universe is God). When we say that God's essence and energies are distinct, we do not mean that God does not convey His true nature and divinity to us; or that there is "God in the world" and "God in Heaven". What we mean is that God's grace sustaining the world is truly God Himself, not just God acting through a created intermediary. In that sense, God is supremely near to us.
- The "hypostases" (or persons) is how the essence and energy are "actualized". If "essence" refers to the common attributes, "person" refers to the unique attributes. "Essence" is the nature itself, and "energy" is the potential realization of this nature, but "hypostasis" is how the essence and energy are actually made manifest and existent. Note also that hypostasis is ontologically before essence - for instance, there is no such thing as "humanity" if there isn't a "human" first to actualize it. Likewise, it is incorrect to speak first of God's essence then of God's persons - first of all, there is one God because there is one Father, not because there is one essence.
Orthodox (together with most Eastern Christians) and Catholics (together with most Western Christians) disagree on two things: energy (Catholics would disagree that there is a real distinction between essence and energy in God) and hypostasis (Orthodox would disagree that the hypostatic property of the Holy Spirit is to proceed from the Father and the Son, and would rather say the Holy Spirit proceeds from the Father alone). But either way, I belive that what I said here is agreeable to both, without getting into those details.
So, "God" isn't the essence. "God" is the Father, and the Son is the "Word of God", and the Holy Spirit is the "Spirit of God". But by virtue of coming forth from God the Father, the Son is co-eternal and co-equal with Him, and so He is also fully and really "God", and likewise, the Holy Spirit is fully and really "God". Indeed, if the Son really has to give us the Father's divinity, He must also be God, and if the Spirit is really to lead us to the Son, He must also be God.
So, "God" is not an essence, but a person. And "Trinity" isn't a monad, and certainly not the name of an essence, but the description of the relationship between the Father, the Son, and the Holy Spirit.
There is a reason that the Orthodox consider the New Testament icon of the Trinity to simply be Jesus Christ. In Him we see the Father and from Him we receive the Holy Spirit.