As you seem to be using the terms, you are correct. However a large part of Christianity does not use these terms in this way and they would respond: ???????
To use your termenology, i.e. "gift of the spirit" and "fruits of the spirit," your comment is correct. The "gift of the Spirit" makes it possible to bear the "fruits of the spirit" which are listed in Gal. 5:22-24.
Just for comparision, the fruit of not having the HOly Spirit is listed in the preceeding verses, starting at verse 19.
However, most Christians thing of the "gifts of the spirit" as special gifts for ministry. I.e. those listed in 1 Corinthians 12 and elsewhere. These equip the saints to go out in service. This would be like mercy, teaching, knowledge, etc.
These gifts are quite different from the "gift of the spirit" that you mentioned that equips us to becoming like Jesus. The ones you mentioned are given in order to edify the person who receives the Holy Spirit. While the "gifts of the spirit" are not to edify the one receiveing them, but to be a blessing to others and to equip one to spread the gospel.
Think of it this way.
I hope that helps.