Tension $T$ is the reactive force in the string when you pull at both ends with force $F$. If the string has stopped stretching or accelerating then $T=F$.
The action-reaction pair of forces in Newton's 3rd Law are two equal but opposite forces with the same cause which act mutually on two different bodies : object A exerts force $F$ on object B, and B exerts the same force $F$ back on A.
So the two forces $F_L$ and $F_R$ which you apply at the left and right ends of the string are not an action-reaction pair : although they might be equal and opposite and both caused by your muscles, they both act on the same object - the string. The paired force to your pull $F_L$ with your left hand on the string is the pull $T=F_L$ of the string back on your left hand. And likewise for your right hand. Only one force in each pair acts on a particular Free Body ; the paired force acts on the paired Free Body.
Tension is transmitted through the string, from one particle to the next, in both directions. In ideal, massless strings tension $T$ is the same at all points between the applied forces $F$ at each end. If you divide the string into two parts (left and right) by an imaginary perpendicular line, creating two Free Bodies, then the force which each Free Body exerts on the other is the tension $T$. These two forces form an action-reaction pair. In real strings (which have mass), the tension can vary along the string - eg if it is being accelerated, or if it is suspended in a gravitational field.
The two applied forces $F_L$ and $F_R$ are not necessarily equal. The resultant (net) force accelerates the string. If the string has zero mass, this means infinite acceleration, so $F_L \ne F_R$ is inconsistent with a massless string.