I get this error message I have attached to this message when I try to add either of the defined projections to a newly created data frame. I have shoen the extent of the two defined layers as well, As can be seen they are the same.

The extent values are far higher than they should be for GCS. So I guess that they must be projectied data. But this also fits with what they have been defined as since both World_equidistant_cylindrical and sphere_erquidistant_cylindrical are projected coordinate systems.

I am confused about what you mean when you say that the extent will not change when you define a new projection. Because at the same time you say that if the two layers were defined different (which they are ) and have the same extent then one or both have been improperly defined.

This to me sounds reverse.