That is the exact description of my noise!I've chased the same noise for months on my 08 EX auto
it's hard to tell from the videos but if my description matches your case then I assume it's the same noise, if that's the case I might have an easy fix for you!!!
from a cold start the engine sounds smooth for like 5 seconds and then the tapping begins! , fades a little when engine is fully warmed up but never completely goes away, and it doesn't get louder with revving or load so it's not a bearing/rod knock
I've done multiple short interval oil changes (5-20 and 5-30) , adjusted the valves twice , removed valve cover and oil pan to check oil pump,chain and guides, chain tensioner, camshafts, balance shafts.... all checked out ok so I guess its just piston slap
Here is the fun part, I noticed that the noise goes away when I drive hard, like really hard!! put here in D3 and let her rip ( that's some real Honda DNA in there ) and after about a year of hard driving the noise became much quieter (just the tapping noise, other engine noises are the same) and continued to quieten even further during the three years I've owned my element, I almost don't hear it now except for a couple minutes on a very cold start.
my theory is that these engines never got broken in properly, they were driven easy at a steady rpms early on especially on the 07+ elements with the 5 speed auto were you can stay around 2k rpm on the highway
Please take it back to the dealer and have them double check everything, make sure it's not an actual engine problem (valves/rods/bearings/chains....) that will get worse if you drive harder. if you rule out everything else maybe give my method a try(at your own risk!)