(1) If the Banach space $X$ is separable; and if you use the Lebesgue-measurable sets on $[0,1]$ not the Borel sets; then all three definitions are equivalent.

But of course the main thing of interest is not "measurable function" but "integrable function". When $X$ is not separable, you probably want the Bochner integral, using the definition pointwise a.e. limit of simple functions. There is also the Pettis integral, using weak measurability, but its properties are much worse than the Bochner integral.

For (2), it is clear with the Bochner definition of measurable: pointwise a.e. limit of simple functions. Again you want a complete measure (like Lebesgue) so that your sequnce can do weird things on a set of measure zero.

Plug:
Edgar, G. A. Measurability in a Banach space. Indiana Univ. Math. J. 26 (1977), no. 4, 663–677.
Edgar, G. A. Measurability in a Banach space. II. Indiana Univ. Math. J. 28 (1979), no. 4, 559–579.

(1) If the Banach space $X$ is separable; and if you use the Lebesgue-measurable sets on $[0,1]$ not the Borel sets; then all three definitions are equivalent.

But of course the main thing of interest is not "measurable function" but "integrable function". When $X$ is not separable, you probably want the Bochner integral, using the definition pointwise a.e. limit of simple functions. There is also the Pettis integral, using weak measurability, but its properties are much worse than the Bochner integral.

For (2), it is clear with the Bochner definition of measurable: pointwise a.e. limit of simple functions. Again you want a complete measure (like Lebesgue) so that your sequnce can do weird things on a set of measure zero.

Plug:

• Edgar, G. A. Measurability in a Banach space. Indiana Univ. Math. J. 26 (1977), no. 4, 663–677. MSN.

• Edgar, G. A. Measurability in a Banach space. II. Indiana Univ. Math. J. 28 (1979), no. 4, 559–579. MSN.