It is not correct. The meaning of escape velocity is defined the initial kinetic energy in which a particle can go to infinite without going back. That is the kinetic energy have to have the same magnitude as the gravitational potential on Earth given by $mv^2/w=GMm/R$. Since the energy is conserved, it does not matter which direction you are pointing to.

For the rocket, it has no initial KE and it gains KE and PE by consuming its fuel. The reason that a rocket move straight up is to reduce air friction at the beginning. Then it follows a slanted path later is to increase flight time so a only a lower efficiency engine is required.

It is not correct. The meaning of escape velocity is defined the initial kinetic energy in which a particle can go to infinite without going back. That is the kinetic energy have to have the same magnitude as the gravitational potential on Earth given by $mv^2/2=GMm/R$. Since the energy is conserved, it does not matter which direction you are pointing to.

For the rocket, it has no initial KE and it gains KE and PE by consuming its fuel. The reason that a rocket move straight up is to reduce air friction at the beginning. Then it follows a slanted path later is to increase flight time so a only a lower efficiency engine is required.