How many different Spirits do Biblical Unitarians believe God has sent or gifted to be in believers? There are many names given in Scripture to a Spirit or Spirits sent to live with and, most especially, live within those who have received the salvation inherent in the Gospel.  This is not an exhaustive list but a representative list only and edited additions are welcome.

And I will ask the Father, and he will give you another Helper, to be with you forever, even the Spirit of truth, whom the world cannot receive, because it neither sees him nor knows him. You know him, for he dwells with you and will be in you. - John 14:16-17


For it is not you who speak, but the Spirit of your Father speaking through you - Matthew 10:20


And because you are sons, God has sent the Spirit of his Son into our hearts, crying, “Abba! Father!” - Galatians 4:6


You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him. - Romans 8:9


Or do you not know that your body is a temple of the Holy Spirit within you, whom you have from God? You are not your own - 1 Corinthians 6:19

Just in these five verses alone we have 7 different names or titles for this &quot;sending&quot;:  Helper, Spirit of truth, Spirit of the Father, Spirit of his son, Spirit of God, Spirit of Christ, and Holy Spirit.
Do Biblical Unitarians understand multiple names for just one Spirit, multiple names for several different Spirits, individual names of individual Spirits, or something else?
A Biblical Unitarian appendix to Biblical commentary Appendix 6: Usages of &quot;Spirit&quot; lists 15 different senses for the term 'spirit' ('pneuma' in the Greek) in the Bible.
However, as Appendix 11: What is the Holy Spirit? says:

&quot;“HOLY SPIRIT” is primarily used in two very different ways: One way
is to refer to God Himself, and the other way is referring to God’s
nature that He gives to people.

So the first sense is a synonym for God. The second sense itself has synonyms. For example, the 'Helper' mentioned above in John 14:16 is another name for this second sense, so 'holy spirit' as a gift.

&quot;18) The gift of holy spirit as “the helper” [...] The “holy spirit”
is identified as a “helper” to believers in John 14:26. [...] The
helper (the holy spirit) is sent by the Father (John 14:16-17) and
Jesus (John 16:7). It does not speak on its own, but it speaks only
what it hears (John 16:13). Thus, the gift of holy spirit is directed
by God and Jesus, which is what we would expect since it is God’s
nature born in us.&quot;

So Biblical Unitarians would understand various uses related to the holy spirit as mapping onto either of the two senses described above, and the second sense of a gift can be directed by either God or Jesus. As examples of both instances of the second sense from your list above, 'Helper', 'Spirit of truth', 'Spirit of your Father', 'Holy Spirit' = gift from God. 'Spirit of his Son' = gift from Jesus.
However, as mentioned, sometimes uses of 'spirit' would map onto other senses as listed in Appendix 6 above.