According to Protestantism is the Holy Spirit a Spirit of a Spirit? God is a Spirit as stated in the Bible. The Holy Spirit is described as the Spirit of God. Does this mean that the Holy Spirit is a Spirit of a Spirit? How does this relationship work?

I know the Holy Spirit is one with the Trinity - God the Father, the Son, and the Holy Spirit. I know the Son came in flesh and the Spirit in Spirit. So God and the Holy Spirit are both Spirits with the latter being a Spirit of a Spirit?
Bible references: 


  1 Cor 12:3 - Wherefore I give you to understand,
  that no man speaking by The Spirit of God calleth Jesus accursed...


and also


  John 4:24 - God is a Spirit: and they that worship him must worship
  him in spirit and in truth.
Both of these verses use the same Greek word: pnuema, but in slightly different contexts. The first uses it as a name, the second as a description of essence. The Spirit of God (1 Cor. 12:3) is the man name used of the third person in the trinity whereas saying that God is a Spirit (Jn. 4:24) places the one God in the essential category of Spirit as opposed to within the category of material creatures or gods which must be worshiped within physical temples. Therefore, blueletterbible.org would categorize the first use under the "I." and the second use would fit definition "III."

An analogy would be the use of the word Son. It is one word but with several meanings and potential definitions. I may say: "I have a son" or "You are my son" or I might call you it as a name "Come over here Son!"