Loud engine sound during first start

when the engine is cold, you get a high rev start, and it gradually drops as it warms up.
all the parts are worn on an old engine and could make more noise than newer.
plus the oil has to circulate a bit for lubrication. but i would hesitate to make any diagnosis without hearing it, and maybe in person, and doubt it would be possible to determine if it could be driven without incident.
other possibilities might be head gasket or semi clutch part between the engine and transmission.
how many miles on her?
it might be helpful to know the trouble codes, and you would need a prius aware scanner for that

 

Engine is really loud if the catalytic converter is missing. If the coupling between that device and the exhaust system is in bad shape it might open up at the junction when cold, and then close up when warm. So loud when cold, quiet when hot.

There are many other things that might make the engine run rough when it is cold, but if the tone is the same, just the volume varies, inspect the exhaust system.

 

sounds like a big risk, unless you don't mind driving that for and turning it down, or if you are ready to take on a project car.

 

I don't hear anything unusual in that video clip.