Is this battery or fuel pump...or what?

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That chap in video is trying to sort out a fuel leek with his filter housing.
You likely have air getting into the system and @miktdish explains the result in post above. Air could be getting in at the filter itself or any connection right up to the engine.
The clear section of fuel pipe on mine I fitted so as be able to track down an air getting in issue on my Tdi.
Had same symptoms as yours, first start slow with some spluttering until system filled and pump self bled.
Ok then for the rest of the day but overnight a slow leak of air would allow fuel to drain back to tank.
With mine issue was a filter connection.
 
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