Why are there differences in the "Greeting" of the Mass? This website lists three options for the greeting recited by the presider of the Mass.
After the following:

Priest:  In the name of the Father, and of the Son, and of the Holy Spirit.  [All together make the sign of the cross.]
All:  Amen.

Then, one of three options:

Option A
Priest:  The grace of our Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all.
All:  And also with you.



Option B
Priest:  The grace and peace of God our Father and the Lord Jesus Christ be with you.
All:  Blessed be God, the Father of our Lord Jesus Christ.
or All:  And also with you.



Option C
Priest:  The Lord be with you.
[or Bishop:  Peace be with you.]
All:  And also with you.


Why are there three options?
Which authority determines which option to use? Is this the presider's choice alone, or does one in authority above the presider make this determination?
Not just the introductory right, there are numerous variation for prayers all through the mass. There are 13 Eucharistic Prayers approved by Holy See for use in the United States.src

Why are there three options?

Because even though all Mass are same in essence, people are invited to meditate upon a certain theme in each mass. Prayers differ depending on the theme of the mass.

Which authority determines which option to use? Is this the presider's choice alone, or does one in authority above the presider make this determination?

Answer is both. Mostly it is the presider's choice. But he has to follow the guidelines in the General Instruction of the Roman Missal Chapter VII. The bishop can make changes but all changes have to be approved by Congregation For Divine Worship And The Discipline of The Sacraments