Trinity as God in Three Persons or Manifestations?

Persona or Hypostasis?

This issue goes back in part to a question of translation from Greek into Latin. The Greek Fathers used the formula “one essence {ousia} and three hypostases." Following the Latin of Tertullian, who was the first writer known to use the term "trinity," the Western Fathers spoke of "one being in three persons." But "persons" is not exactly the same as "hypostases."

As St. Gregory the Nazianzus (d. 389) explained:

We use in an orthodox sense the terms “one essence and three hypostases”, the one to denote the nature of the Godhead, the other the properties of the Three; the Italians mean the same, but, owing to the scantiness of their vocabulary, and its poverty of terms, they are unable to distinguish between essence and hypostases, and therefore introduce the term “persons”, to avoid being understood to assert three essences.

So in fact, "hypostasis" may be closer to "essence" than "person." Indeed, in philosophy, hypostasis is defined as "the substance, essence, or underlying reality". But it would hardly do to speak of "one being in three substances," because "substance" was already in use as the translation of "ousia." St. Gregory's complaint was not without basis.

Scriptural basis

Several scriptural verses are as cited for the idea of each of the three each being a "person." Besides those that simply mention the Father, Son and/or Holy Spirit, the following is often seen as evidence for each of the three having personhood.

The grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit, be with you all. (2 Corinthians 13:14)

These show the three "persons" to have distinct functions. However, it should probably be admitted that it would be easier to find a scriptural basis for Gregory of Nazianzus' formula, where "hypostasis" refers to the properties or characteristics of the three, rather than their personhood in the sense that modern English understands the term.

Manifestations?

The OP asks about Manifestations as a possible alternative for Persons. This would certainly be objected to on the grounds that it is too close to Modalism.

Modalism, also called Sabellianism, is the unorthodox belief that God is one person who has revealed himself in three forms or modes in contrast to the Trinitarian doctrine where God is one being eternally existing in three persons.

The current Greek and Latin formulae to describe the Trinity may not be written in as solid, as the Ten Commandments, but they would be practically impossible to change. Knowing that the Greek word translated as "person" was understood less definitely in ancient times helps clear the matter up. "Persona" doesn't quite capture the subtlety of "hypostasis."

In my original answer I neglected to deal with the secondary question: "Where in scripture is it essential-for-salvation to claim God is in Three Persons?"

First a disclaimer: my personal view is that it not necessary. But scriptural support for this doctrine is found in the trinitarian baptismal formula.

Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit. - Matthew 28:19

Since most churches affirm that baptism is necessary for salvation, this scripture supports the claim that trinitarian belief is also necessary. However, IMO this argument fails because in Acts and Paul, baptism is performed in the name of Jesus. Moreover Apollos became a significant figure in the early church even though he "knew only the baptism of John," which used neither the trinitarian formula nor the name of Jesus. (Acts 18:24-27)

Nevertheless, the formula given in Mt. 28 indeed provides support for the need to believe in trinitarianism, since Jesus commanded it.

(again these arguments are not my personal views; the are my answer to one of the OP questions)

Regarding the question about reconciling the OP verses, there are various was to attempt them. One is to affirm that the Holy Spirit described in trinitarian theology is not the same as what is mentioned in and Paul. In the OT, the "spirit of the Lord" is simply God, not the third person of the Trinity and Paul, is simply using "Spirit" in that sense. Paul's reference to the Jesus as a life-giving spirit similarly does not refer to the Holy Spirit but to Jesus' own life-giving power.

Elsewhere Paul refers to the Holy Spirit differently, with a clear distinction between God and the Holy Spirit:

God did not call us to impurity but in holiness. Therefore whoever rejects this rejects not human authority but God, who also gives his Holy Spirit to you. - 1 Thess. 4:7-8