It is true that the NT speaks of the "spirit of man" (eg, 1 Cor 2:11) as distinct from the "Spirit of God" (eg, 1 Cor 2:11, Rom 8:9, etc).
However, when the word "spirit" (πνεῦμα) is accompanied by the adjective "holy" (ἅγιος) it ALWAYTS refers to the Spirit of God, which is the also the Spirit of Christ, Rom 8:9.
It is this simple fact that leads almost all translators to capitalize "Holy Spirit" wherever it appears.
Put another way, "Holy Spirit" does not ever refer to the spirit of man/humans.
Article
The rules covering the use of the Greek article are different in English and Greek. For example, in English, when describing a person in the third person using a title, the article ("the") is required. [Eg, Go and tell the sergeant ... ] This is untrue in Greek.
In Greek, the first mention of a noun in any context does NOT use the article, but subsequent mentions that refer back to this first mention, must use the article. Grammatically, this means the Greek article in such contexts is "anaphoric".
This can be clearly seen in Luke 2:25-35 describing the prophecy of Simeon:
- the first mention "Holy Spirit", V25, has no article
- the second mention "The Holy Spirit", V26, has the article
- the third mention "The Spirit", V27, also has the article
The second two refer back to the first and confirm that Luke is discussing "The Holy Spirit", because English always requires an article for the title of a person.
This is further confirmed by the fact that V26 says that Simeon was "divinely inspired"
APPENDIX - Personal Names
The reverse is also true - English NEVER uses an article before a person's name in the vocative sense. [Eg, we never say, Go and tell the John that ... ]
However, in Greek, personal names often require the article especially when they refer back to a first instance in a passage.
Therefore, sometimes English must delete the article when the Greek has it, and sometimes English must supply the article when the Greek has none. That is, the rules around the definite article are different in each language.