Ah, I would have thought that they are the same thing. A counter is an integer - as in a number. In this case, the space allocated for it as an OID is 64 bits worth (which is important since you want adequate space to take into account rollover!).
Concerning what you have shown, you are referencing two different mechanisms as to how things are defined. The top one is pulling things from the MIB itself, which, as you point out, has the concept of Strings, Octets, Counter32 and Counter64, etc.
However, the bottom is specifying how attributes are stored in the actual Spectrum database (as opposed to the MIB) - in this case as an integer since we are dealing with numbers. The other types include Booleans and Date/Time. The mappings in Spectrum for this sort of thing do NOT tie into how things are defined in the MIB - except when dealing with the obvious - as in you aren't going to be mapping a String to an Integer!
If you want to start looking into Attribute Types mapped into models then you can use the MTE for this sort of thing. However, I would express caution that if you don't know how to use the MTE then you can right royally screw up the models on the system. So, make sure you take a backup of the SSDb beforehand. Also, Spectrum needs to be fully down before you perform any MTE changes (NOT that I am suggesting you do this at all here).
However, before you even consider doing any of this, I would ask as to why is this causing you issues? It seems that you are making a problem out of something that isn't a problem.