Let’s break these down so that the jargon is easier to understand.
Cremation container/Alternative container/ Cremation casket: These terms refer to the container that is used to place the deceased into the cremation chamber before the cremation process.  The body needs to be contained in a rigid combustible container for the safety of the crematory operator, so that it can be easily stored, lifted, and moved. It must also create a barrier and contain bodily fluids. Stop for a moment to consider what things would be like without them – unsafe and undignified.
The cremation container/alternative container are terms that can be used interchangeably, and they refer to a lower cost container with the specific design of being used for the cremation process. From a legal standpoint, the “minimum alternative container” is exactly what it says.  It is required to be offered by federal regulations in the FTC Funeral Rule.
A cremation casket is a container that is like a burial casket in appearance and is designed for the presentation of the deceased for a viewing or visitation. This type of casket contains minimal metal parts for the cremation process.
The temporary container and urn perform basically the same function – they contain the remains after the cremation process. This is the “ashes box.”  A temporary container is almost always a plastic container with a plastic bag containing the remains inside.  A “permanent” urn is typically made from metal, stone or wood. These are used for permanent holding of remains, whether it be in a home, a columbarium or for in ground burial.