Given an irreducible curve $C$ of degree $d$ in $\mathbb{P}^r$ and a general hyperplane $H\subset\mathbb{P}^r$, the uniform position theorem states that any $r$ points on the hyperplane section $H\cap C$ will be linearly independent. (Suppose we are working over the complex numbers) One reference I have found so far is Arbarello-Cornalba-Griffiths-Harris, Geometry of Algebraic Curves I.

Now I wonder, what general really means in this statement:

My first hope is that any transversal intersection will do. Is this true? My feeling is that this might not be good enough.

Should it fail, suppose I am given a subspace of linear forms on $\mathbb{P}^r$ that induces a base point free linear system on the given curve $C$. Can I find a general hyperplane section in this linear system such that the uniform position property holds?