Eddy current in transformer and its minimization
No, eddy current loss is not the same as core loss.  Core loss usually refers to the material itself absorbing some of the magnetic field energy as it is magentized and demagnetized.  Ideal material acts like a ideal spring, in that you get all the magnetic field energy back that you put in.  Real materials will be somewhat lossy, just like real springs.

Eddy currents are currents caused by the changing magnetic field.  This is exactly the effect that causes currents in the secondary when the magnetic field is changed due to how the primary is driven.  Think about the difference between a conductive metal core and another secondary.  The problem is that there basically isn't one.  A conductive core acts like another secondary that is always shorted.

So does that mean you can never use conductive material in a transformer core?  Not completely.  At first glance, you use something like ferrite that has reasonable magnetic properties but does not conduct.  However, unfortunately materials with really good magnetic properties are conductive, like iron.  A iron core transformer will be significantly smaller than one that handles the same power but has a ferrite core.

Therefore it's worth getting clever about how to use iron but keep it from conducting to make a shorted secondary.  One way is what's known as a powdered core.  Iron is used ground into small pieces, which are suspended in something that insulates.  You still get much of the iron properties, but the bulk material can't conduct much because each of the individual iron particles are largely insulated from each other.

Other more common methods of using conductive material like iron are based on the observation that we only need to prevent the material from conducting in a particular direction, which is circularly around the center of the core.  One way to do this is to make the core out of a bunch of thin iron plates that have a even thinner insulating layer between them.  Tiny eddy currents still exist, but only within each thin sheet, so are greatly reduced.  This is often called a laminated core, and is quite common.  Take a look at a large line-frequency power transformer, and you will probably be able to see the stackup of plates.

Another method is to start with a long thin sheet of iron and wind it up to make a bulk volume, with a thin insulating layer between each layer of winding.  Now the transformer core looks like a secondary winding, but since the ends aren't connected, it is always open circuit and therefore not present electrically.  That's often called a wound core.  Again, look at large line-frequency power transformers and you will sometimes be able to see a spiral of thin layers, instead of parallel flat layers as in a laminated core.