please HELP! how can I create this specific smoke effect

That looks to me like Turbulent Displace and / or Vector Blur.

Looking at my screenshot:

• I precomposed the logo into comp - so that effects would extend beyond the edges of the images.
• At the top of the comp is just the logo precomp, so that it matches the video you linked.
• The second comp has CC Vector Blur and Turbulent Displace. 
• CC Vector Blur uses the 3rd layer as a vector map to drive the "warping"
• The third layer has Fractal Noise on it, with the offset and evolution keyframed.  
• The Turbuluent Displace also have the offset and evolution keyframed.  These keyframes all help create a flame like distortion.