Tough to diagnos a written sound but thinking about it, upon hot or cold start would not think it would have anything to do with something loose (piston slap) and metal heating up expanding and then quieting down.
Sometimes it's tough to dicern valve gear noise (I call rapping) from knocking. Sounds as though what you are describing could be oil preasure related as temperature seems to have nothing to do with it.
I do notice on my car that I get some valve train noise upon start up when I am down a little on oil. Keeping full tends to keep noise away at start up.
Would be easier to guess if it happened when cold.
Sorry can't be of more help,