There is more fashion than science in this issue. The nitpicking, obsessive-compulsive focus on the distinction between speed and velocity in physics is a relatively modern development (less than 50 years??). read more
So escape velocity is just a speed of an object which can escape from a heavy body's gravitational field without any external force. But the object will not be able to escape that field if it will travel towards the center of the body. This means escape velocity have a sense of direction. That's why we call it a velocity. read more
If I place a rock, say, near the Earth, with a velocity high enough to escape the Earth's gravity. It will only have acceleration based on gravity; as it is unpowered, but whether or not it escapes will depend entirely on whether its velocity is above the escape velocity for the position it is in. read more
Escape speed is the minimum speed required for the object to just reach infinity. What this means is that the kinetic energy of the object during launch is just sufficient for the object to come to a rest as it approaches infinity. read more