“I observed in the tract on the Holy Spirit the equation: ‘Jesus Christ = The Lord; the Lord = That Spirit, Therefore: Jesus Christ = That Spirit.’ It also continues to prove that Jesus is our Comforter, therefore Jesus is the Holy Spirit. This assertion can be misleading as it gives the impression that the Holy Spirit is Jesus. I am sure that’s not exactly what you mean (or is it?). After all, it is quite clear that ‘the Holy Spirit’ is a ‘third’ ‘entity’ though not a person. (Matthew 12:30-32; Matthew 28:19, etc.) I have not distributed this tract as it can misrepresent the truth as we know it. Please consider and clarify.”
It is true that the Lord Jesus is that Spirit, our Comforter, but to state it more completely, the Holy Spirit is the Spirit of the Father coming to us through His Son, giving us the added benefit of having the Spirit of His Son who was victorious over all sin. (Galatians 4:6) When speaking of the promised Comforter Jesus said,
And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever; Even the Spirit of truth; whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him; for he dwelleth with you, and shall be in you. I will not leave you comfortless: I will come to you. (John 14:16-18)
Here Jesus made it clear that He would return to His disciples as the promised Comforter. He said that this Comforter was already dwelling with them, on the outside, but soon He would be in them. Then Christ said that He would come to them. Jesus continued, “At that day ye shall know that I am in my Father, and ye in me, and I in you.” (John 14:20) Jesus said that the day the Comforter comes to them, His disciples would know that He was in them. He further explains, “If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him.” (John 14:23) Here we see that the promised Comforter that would come to the disciples was both the Spirit of the Father and the Spirit of His Son. When we receive the Spirit of the Son, we receive the Spirit of the Father because the Father dwells in His Son by His Spirit.
The Holy Spirit proceeds “from the Father” (John 15:26), to the Son, and from the Son to us. (See John 3:34 and Titus 3:5, 6.) Then we can have personal fellowship with both the Father and His Son. “Truly our fellowship is with the Father, and with his Son Jesus Christ.” (1 John 1:3) “He that acknowledgeth the Son hath the Father also.” (1 John 2:23)
You wrote, “it is quite clear that ‘the Holy Spirit’ is a ‘third’ ‘entity’ though not a person.” Actually, this is not clear, it is not even taught in the Bible at all. According to Webster’s New Explorer Dictionary, the word “entity” means, “something with separate and real existence.” This could not be true of the Holy Spirit, for the Bible says that it “proceedeth from the Father.” (John 15:26) The Greek word that was translated proceedeth is in present tense, indicating a continual process of proceeding from the Father. Since the Holy Spirit continually proceeds from the Father, it could not have a separate existence from the Father.
I hope this helps to answer your question.