What is the name of the printing process that can be used to print directly on plastic cosmetic packaging or a glass jar?
It can be a printer with metal or colour. And it is not a clear label.
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There's a strong likelihood that those items were printed using a process called rubber pad printing. It uses a (typically) silicone rubber bulb of sufficient size and shape to hold the desired image, when the following characteristics are considered.
The shape of the bulb must take into account distortion when compressed against the print source (think old-style movable type, or similar raised engraving), and also when compressed against the target.
There are videos showing multiple color machines in operation. A single color operation involves the pad/bulb contacting the previously inked engraving, transferring the ink/paint to the bulb. The bulb then moves to the item and is again compressed against the surface, imprinting it.
Using "rubber pad printing" as the search terms, I found that this method is used for printing on writing instruments such as pens, on bottles, as you've shown and any suitable non-flat surface. Flat surfaces are often silk-screened, which requires less equipment, or less expensive equipment, although that is a generality and subject to exceptions.
The photo below shows that flat surfaces can be printed too, which is unsurprising.
It can also be made in flexo printing. specifically the bottles have transparent stickers on to it. In paper or carton either they use offset or flexo.
AS far as I know, it's called Pad printing or Tampography. It also can be screenprinted to a flexible surface as plastic film and then transfered to the shaped packaging.