In New Jersey, any plate issued since 1959 is still potentially valid, and can be used on the road if it has been continuously registered to a vehicle since it was first issued.
But sometimes plates will get lost, or become damaged, or a vehicle owner will simply want to get their existing plate number on the latest design, like replacing 1959-base tan plates with 1979-base blue ones, or replacing embossed plates with new flat ones.
Replacement plates can be obtained from the MVC - the new plates will have the same serial number as the originals, and they will also have the separator (previously a dash, now the state map outline).

The important point here for classifying remake plates is the presence (or absence) of the separator.

If a new plate with an old number has the separator, it is a regular "replacement remake" of a live plate number.   These can be seen on this page, below.
(If it does not have a separator, it is what we will now classify as a "vanity remake" - see this page for details.)

Below are examples of live plates which were remade onto new baseplate designs - a.k.a. replacement remakes.
(I have limited the display to a few of each type on each base, and shown the original plate wherever possible.)

I put a 7-digit remake plate here because it's impossible to tell where it belongs (no separator!).

Many thanks to Steve Bier for investigating and helping to sort this out.