Most common name for function F in Finsler geometry

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In Finsler geometry, a Finsler manifold $M$ is a differentiable manifold, together with a function $F$ defined on the tangent bundle of $M$, that is an asymmetric norm on each tangent space. I have found many different names for this $F$, such as Finsler function, Finsler norm, Finsler metric or Finsler structure. Basically, I would like to know which name I should use. Are there any well-known references that I could follow?