I take the barrel out by putting it on the battery and shimmying the tube off of it and I haven't had an issue yet. I will give you a step by step of what I do but honestly, I would suggest only taking the mouth piece silicone out and the inner silicone seal and running under hot water until you see the
juice cleaned around the wicks. When you put the inner seal back in, make sure it is wet and if you have both clearos and CE3s, use the plastic tip of the clearo to push the silicone down - it will seat it pretty well and give you a good seal. It works well to clean them. Taking them completely apart is really only needed if you plan to dry burn which I have had about a 50/50 success rate with......some work, some don't. I got about two months out of the CE3 that way before ever taking them completely apart.
The easiest way is to screw the CE3 on to the battery and as I said, shimmy the tube off gently!! The center metal tube should wiggle out and the silicone can be pushed out of the tube. I assume that is what the video did too.
For the Clearos, I do the same thing. Run them under hot water and I soak them as well but I havent personally had great success cleaning the clearos or dry burning them and getting them to work well afterwards. It really pushed me away from the clearos. I have heard others say a vodka soak and a dry burn equals like new but my results varied.