Listening and re-listening to the video clips you posted, it sure sounds like lifter tick to me. Since it goes away after 3-4 operations of that valve, it isn't anything serious. When it starts taking several operations of the valve, or taking several seconds to go away, plan a replacement party. Simply changing to a higher detergent oil can sometimes help if the hydraulic element is clogged with sludge or varnish. Don't expect miracles, but some folks have had luck with it, though it takes about 1000 miles and some highway time to work its way through.

Tensioner problems would be causing chain slap, which is a different noise than what you have. I am curious about the whining noise and difficult start in the audio-only clip though...

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]