How different are these terms: Phylloclade, Phyllode, Cladophyll and Cladode?
Apart from the discrepancy of the cladophyll term I can say that both your teacher and the wiki page are correct.

If you read the Wiki page carefully it states that


  Phylloclades and cladodes are flattened, photosynthetic shoots, which are usually considered to be modified branches. The two terms are used either differently or interchangeably by different authors.


(As you have also quoted) 

This is simple because both phylloclades and cladodes are stem modification for photosynthesis and xerophytic adaptation as well. So in this context they are interchangeable

Also if you have further read the article it states why the terms are interchangeable and why not. The different levels of definition are given:


  Definitions of the terms "phylloclade" and "cladode" vary. All agree that they are flattened structures that are photosynthetic and resemble leaf-like branches. 
  
  
  
  one definition, phylloclades are a subset of cladodes, namely those that greatly resemble or perform the function of leaves,[2] as in Butcher's broom (Ruscus aculeatus) as well as Phyllanthus and some Asparagus species.


The definition given by your teacher can be interpreted by these two definitions stated in the Wiki page:


  By an alternative definition, cladodes are distinguished by their limited growth and that they involve only one or two internodes.[3] By this definition, some of the most leaf-like structures are cladodes, rather than phylloclades. By that definition, Phyllanthus has phylloclades, but Ruscus and Asparagus have cladodes.
  
  
  
  Another definition uses "phylloclade" to refer a portion of a leaf-like stem or branch with multiple nodes and internodes, and "cladode" for a single internode of a phylloclade.


But phyllode is a leaf modification, specifically a modification of the petiole for photosynthesis and a xerophytic adaptation to some extent. (Source is the Wiki page itself)

So to conclude 


phyllode is completely different from phylloclade and cladodes.
Taking the type of morphology of the plant into consideration and the definition used, phylloclade and cladodes may be used interchangeably or may not be.