Jesus did not say that the Spirit would not talk about Himself. Neither did Jesus say that the Spirit would always only glorify Christ and never glorify Himself.
The Spirit has spoken about Himself many times in Scripture, and He has glorified Himself in many ways in Scripture.
Seeing how the Holy Spirit is not really speaking but speaks what He hears, such references should be understood in that way for how the prophets spoke & for how the scripture comes about when those words were not coming directly from Himself. When the disciples speak or had written, they were inspired by the Holy Ghost to signify how they were speaking & writing as in.. not their own words but by the Spirit of God. That said.. Spirit of Someone testifies the words yet still, coming from Someone Else, even if the Holy Spirit is God, as what He speaks are not His own words, but words from Another & so the gory of those words from Another.. goes to the "Another".
Jesus did say what defines a false witness from a true witness, even for Himself.
John 5:31 If I bear witness of myself, my witness is not true. KJV
So that also applies to how the Holy Spirit witnesses thru us.
John 15:26 But when
the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father,
he shall testify of me: 27 And ye also shall bear witness, because ye have been with me from the beginning. KJV
Note what a disciple of Jesus Christ is supposed to do when bearing witness of Jesus Christ as the example was set by John the Baptist on witnessing in preparing the way for Him as being that prophesied voice in the wilderness.
John 3:28 Ye yourselves bear me witness, that I said, I am not the Christ, but that I am sent before him. 29 He that hath the bride is the bridegroom: but the friend of the bridegroom, which standeth and heareth him, rejoiceth greatly because of the bridegroom's voice: this my joy therefore is fulfilled. 30 He must increase, but I must decrease. ~ KJV
So a true witness is not to speak of himself in seeking his own glory since the witness is to decrease so that He may increase. How is that rightly dividing the word of truth? Jesus said so plainly here in regards to how men witnesses.
John 7:18 He that speaketh of himself seeketh his own glory: but he that seeketh his glory that sent him, the same is true, and no unrighteousness is in him. KJV
There is no unrighteousness within God. The Holy Spirit has been sent to do a job of withnessing of the Son in seeking His glory so how can He do that except thru us?
When you are witnessing by yourself or worshiping God by yourself, the only way your witness in ministry & worship can be true if the Spirit's witnessing with you for why those who are led by the Spirit are to honor the Son in worship if they wish to honor the Father at the same time. That is the Holy Spirit fulfilling His role for what He has been and still is sent for in bearing witness of the Son in seeking His glory.
John 8: 17 It is also written in your law, that the testimony of two men is true. KJV
John 16:13 Howbeit when he, the Spirit of truth, is come, he will guide you into all truth:
for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: and he will shew you things to come.
14 He shall glorify me: for he shall receive of mine, and shall shew it unto you. 15 All things that the Father hath are mine: therefore said I, that he shall take of mine, and shall shew it unto you. ~ KJV
At one time I had thought that verse 13 in the bold was in regards to how the Holy Spirit witnesses not about Himself, but the Lord corrected me in that He was meaning how the Holy Spirit speak as in how He does not speak His own words as coming directly from Himself, and that verses 14-15 is about how He ministers per the Head of the Church, the body of believers for why everything the Holy Spirit does, the credit & glory goes to Jesus Christ.
That means all the words of the Spirit, all the gifts of the Spirit, & all the fruits of the Spirit are ministered with the understanding of the scripture as a whole in aligning all the truths in His words that they come from Jesus Christ for why Jesus gets the credit & glory for all the Holy Spirit does which automatically goes to the glory of God the Father.
So the Holy Spirit in us is not going to lead believers to testify of Himself in seeking His own glory. That false notion was introduced to the churches by the modified Nicene creed of 381 A.D. that hyped a lot of titles on the Holy Spirit which should have been given to the Son before introducing the practice of worshiping the Holy Spirit with the Father & the Son.
There are no scripture testifying to that is how God wants us to practice in that way, but there are plenty of scripture focusing on the Son in worship for how God the Father wants to be glorified by in approaching Him in worship, anywhere, by.
John 14:6 Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me. KJV
John 10:1Verily, verily, I say unto you, He that entereth not by the door into the sheepfold, but climbeth up some other way, the same is a thief and a robber. KJV
John 5:22 For the Father judgeth no man, but hath committed all judgment unto the Son: 23 That all men should honour the Son, even as they honour the Father. He that honoureth not the Son honoureth not the Father which hath sent him. KJV
There are no scripture opposing the plainness of meaning here for why the Holy Spirit is not mentioned as another way to come to God the Father by or as another way to honor the Father in worship by. John 14:6, John 10:1 & John 5:22-23 is why there are no scripture teaching the practice of worshiping the Holy Spirit with the Father & the Son even though He is God also and that is because the Holy Spirit is to serve as a perfect Witness thru us in testifying of the Son in seeking His glory.