Your difficulty is not that odd since there's some ambiguity in the nomenclature.
So-called "Current feedback amps" connected in the conventional fashion use voltage in and voltage out in the circuits they are part of, and look on paper very much like other op amps. But the difference is that the inverting input is very LOW impedance (ideally, zero), while the two inputs of a traditional op amp are both relatively high Z.
Amplifiers come in at least four different flavors, ideally. Any real amp can be a mixture of them, but is usually trying to be as close to one of the ideals as possible.
The most familiar ideal is the voltage in/voltage out amp. The ideal is infinite input Z, zero output Z, and some voltage gain, maybe a lot.
There are also the transconductance amps: voltage in, current out. The ideal is infinite input Z and infinite output Z. The "gain" is delta current out/delta voltage in.
Then we have current amps: current in and current out, with zero input Z and infinite output Z. The gain is in delta current out/delta current in.
Finally in the last quadrant is a transresistance amp: current in, voltage out, zero input Z, zero output Z. The gain is delta voltage out/delta current in.
We can choose to use current as our signal variable if we like, and sometimes there are advantages. Where, with voltage as the variable we can send the same signal to a number of inputs, with current as the variable we have to replicate the currnt somehow for multiple inputs, or divide it with the associated losses, or have a way of recycling it somehow. This is usually less convenient than feeding a voltage source to multiple inputs. We can use current as the feedback variable as well, and so have a "current-feedback" system.
So far all that is described applies to amps with a single input and a single output. So we haven't even gotten to an op amp yet.
Conventional op amps are a two-input affair with high Z at both inputs; the output is a voltage that is the highly amplified difference of the two input voltages. If this differential gain is high and falls off in a nice way with frequency then we can apply feedback in various ways and get a smaller but more precise gain, or any number of other overall circuit transfer functions and behaviors.
All of the other flavors of amps described can be extended to two-differential-input forms as well, and that generates a bunch more ideal cases. And then we can mix the types of input too: our "current-feedback amp" has a voltage-input noninverting input, a current-input inverting input, and a voltage output. So there are quite a number of possibilites!
If that weren't enough, there are also differential output amps as well.
It would be interesting to see a graphic of all the possible ideal types of amplifier. But then, I don't get out much.