And speak unto him, saying, Thus speaketh the LORD of hosts, saying, Behold the man whose name is The BRANCH; and he shall grow up out of his place, and he shall build the temple of the LORD: Even he shall build the temple of the LORD; and he shall bear the glory, and shall sit and rule upon his throne; and he shall be a priest upon his throne: and the counsel of peace shall be between them both. 
Zecheriah 6:12, 13

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The “temple of the Lord” refers to the heavenly sanctuary where Christ intercedes before the Father.  He is “priest upon his throne”. So the “counsel of peace” refers Christ’s intercession between the Father. Notice what Ellen White has to say about this verse:

“The work of Christ as man's intercessor is presented in that beautiful prophecy of Zechariah concerning Him “whose name is the Branch.” Says the prophet: “He shall build the temple of the Lord; and He shall bear the glory, and shall sit and rule upon His [the Father's] throne; and He shall be a priest upon His throne: and the counsel of peace shall be between Them both.” Zechariah 6:12, 13. GC 415.3

“He shall build the temple of the Lord.” By His sacrifice and mediation Christ is both the foundation and the builder of the church of God. The apostle Paul points to Him as “the chief Cornerstone; in whom all the building fitly framed together groweth into an holy temple in the Lord: in whom ye also,” he says, “are builded together for an habitation of God through the Spirit.” Ephesians 2:20-22. GC 416.1

“He shall bear the glory.” To Christ belongs the glory of redemption for the fallen race. Through the eternal ages, the song of the ransomed ones will be: “Unto Him that loved us, and washed us from our sins in His own blood, ... to Him be glory and dominion for ever and ever.” Revelation 1:5, 6. GC 416.2

He “shall sit and rule upon His throne; and He shall be a priest upon His throne.” Not now “upon the throne of His glory;” the kingdom of glory has not yet been ushered in. Not until His work as a mediator shall be ended will God “give unto Him the throne of His father David,” a kingdom of which “there shall be no end.” Luke 1:32, 33. As a priest, Christ is now set down with the Father in His throne. Revelation 3:21. Upon the throne with the eternal, self-existent One is He who “hath borne our griefs, and carried our sorrows,” who “was in all points tempted like as we are, yet without sin,” that He might be “able to succor them that are tempted.” “If any man sin, we have an advocate with the Father.” Isaiah 53:4; Hebrews 4:15; 2:18; 1 John 2:1. His intercession is that of a pierced and broken body, of a spotless life. The wounded hands, the pierced side, the marred feet, plead for fallen man, whose redemption was purchased at such infinite cost.” GC 416.3

“And the counsel of peace shall be between Them both.” The love of the Father, no less than of the Son, is the fountain of salvation for the lost race. Said Jesus to His disciples before He went away: “I say not unto you, that I will pray the Father for you: for the Father Himself loveth you.” John 16:26, 27. God was “in Christ, reconciling the world unto Himself.” 2 Corinthians 5:19. And in the ministration in the sanctuary above, “the counsel of peace shall be between Them both.” “God so loved the world, that He gave His only-begotten Son, that whosoever believeth in Him should not perish, but have everlasting life.” John 3:16. GC 416.4   

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The Holy Spirit does not intercede before the Father. The Holy Spirit has a unique work which is very different from Christ. Notice what Ellen White says:

“Christ, our Mediator, and the Holy Spirit are constantly interceding in man's behalf, but the Spirit pleads not for us as does Christ, who presents His blood, shed from the foundation of the world; the Spirit works upon our hearts, drawing out prayers and penitence, praise and thanksgiving. The gratitude which flows from our lips is the result of the Spirit's striking the cords of the soul in holy memories, awakening the music of the heart.”  Selected Messages Vol. 1 p. 344.1

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So the “counsel of peace” is between the Father and Christ: this refers to the ministry of Christ before the Father in the heavenly sanctuary. This is very different to the work of the Holy Spirit.

So Zechariah 6:13 does not exclude the Holy Spirit as a Third Person of the Godhead. It is just highlighting Christ’s special work as our High Priest with the Father.