Agreed it certainly sounds tappety on that first start up and the misfire can be seen, engine movement and heard.
A leaking manifold gasket won't make that noise, which is definitely mechanical.
Read the code which will possibly be shown as a misfire on cylinder X due to the problem follower, that you say the noise is louder on the inlet manifold side suggests the possibility that the culprit is on the inlet camshaft side, it's a little difficult when listening on videos but at first start up all the followers sound as if they are taking a while to jack up.
If you have access to and are able to do an oil pressure test from cold this will show if the oil pressure is, or isn't OK, bearing in mind that the oil pressure light only comes on when pressure is as low as 5psi to 10psi on average.
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