When the string player causes small pitch fluctuations by rocking the left hand while pressing the string down, it is called: ?

I think you are talking about vibrato. Vibrato- creating vibration in pitch or volume of a note. String players use their left hand to make these vibrations. Example: cellists usually use arm vibrato which is where the rock not only hand but also their arm to make the vibrations (violins usually only rock their hand).

I hope this helps!