You might observe that the way Euclid uses postulate 1 is to provide geometric constructions of shapes. If you want to argue about what he meant, what he meant was "you have a straightedge". Similarly, postulate 3 says "you have a compass", and postulate 2 says... "you have a straightedge".
You've described drawing several parallel lines that meet at two opposite points of a sphere. Postulate 1 has no problem with that - all of those lines are straight. Postulate 5 has a problem with it, because they meet.
You might observe that the way Euclid uses postulate 1 is to provide geometric constructions of shapes. If you want to argue about what he meant, what he meant was "you have a straightedge". Similarly, postulate 3 says "you have a compass", and postulate 2 says... "you have a straightedge".
You've described drawing several parallel lines that meet at two opposite points of a sphere. Postulate 1 has no problem with that - all of those lines are straight. Postulate 5 has a problem with it, because they meet.