Look around seconds 2-5 in the first video. That looked like a run walk. You pushed him out of it and he started to pace. He may be one of those who will pace until you actively ask for a run walk, depending on if he was set in his gait at one time. To ask for the run walk with most horses, sit deep and drive with your seat. Don't kick; just squeeze a little. Keep your hands low and have light contact with the reins. If you feel a pace, half halt and restart. Half halts are your best friend.
Learn to identify the gaits. Can you tell the difference between a pace and a run walk? The head isn't overly reliable; it better to feel it with your seat. The run walk will move you back and forth (just like a normal walk on a normal horse). Look at the front of your saddle. It is moving back and forth or side to side? Side to side means he's pacing. Also, if you ride on a hard surface, you will hear four EVEN beats if he's running. One, two, three, four. If he's pacing, you'll hear something like One, pause, two, pause, one, pause, two.
If he continues to pace, people wiser than me can help. ;) My rehab pacer is coming along, with only because they helped me out.