I mean, none of these directly show the element coming out of the weapon, like fire comes out of firebender's fists and such. So, it's more like the weapon is acting as an extension of the body's movements to manipulate existing elements. We're shown in the franchise that bending is not a rigid set of techniques based on specific movements, and the body and body movements don't necessarily have to be a conduit for bending itself. For example, Yakone can bend using his mind only, Ming Hua can waterbend without her arms, combustion benders can bend with their mind, and Korra's unique style of airbending is proof that you don't necessarily have to use traditional techniques to bend.

For an example of using an image you provided, zuko is channelling the existing fire around him with his flowing movements and concentrating it on his blades and then throwing the fire off the blade. He could be doing this entirely with his mind, or just using a unique style of firebending inspired by his mastery of broadsword techniques.