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why in this video pilot chute collapses for a while and then inflates again ?( seconds 11 to 13 )

It's not collapsing. It's just pulsing a bit (as 2D pilot chutes do) while it stabilizes, fills with air, and starts to pull.

The slow-mo, I think, makes it appear more dramatic than it actually is.

We have video of all styles of PC exhibiting this behavior. Fabric does funny things (especially in slow-mo) in messy air.

How much of that, do you think is just normal recoil of the fibers in the PC fabric? (vs. something else like turbulent air flow or loading of the center line)

Honestly not sure. Modern PCs are built either with multiple gores on the block, and/or with tapes on the bias in order to reduce pulsing/stretching in the fabric...but I'm not sure if it's possible to ever break down all of the contributing factors accurately enough to assign blame in that regard.

All PC designs do (different) funny things. It's a matter of deciding which behavior makes you the least uncomfortable and rolling with it!

It's not collapsing... is just pulling your pins out... perhpas too much tight (loops), but it is a very slow motion to think so. So nothing is wrong...

And personnally, I will not pack (handle) my pilote chute like you. I prefer to start packing handle PC as mushroom, with mesh in the hand with starting from connexion on bridle between thumb and index (bridle coming from hand front). And so you just have to open your hand to leave the PC with a little move outsite of your body. You can flex the top fabric to compact all in your hand, if you want to make a long delay . In your case, it is not a delay enought to pack PC compact (only necessary to not be unstable by relative wind (handle SU or 4s SD).

I agree with flyjeronimo, looks like it is when the pins are being pulled, maybe they are too tight

It could be that there is a force pulling on the bridle so the PC is inflated, the pins pop leading to temporary slack / lack of force (the "collapsing"), and then the bridle is reloaded once it's pulling on the canopy

I don't think "too tight pins" are a real thing.
The difference between the force needed to pull a "loose" pin and a "tight" pin is really nothing compares to the traction power of a PC.
Test is with a dynamometer and I think you'll be surprised.

About the PC behaviour I think it's a normal thing that can happend, my guess is that with a vented PC this would happend less, but it's just a guess, I never tested it.

If that is what you are going with I would suspect it is due to the additional bridle being released from the container, not "Tight Pins".
The rapid introduction of the additional slack into the system mid extraction would have to have some effect on the PC.