When I measure a physical quantity, like voltage, current, or resistance, or flow rate, volume, kilometers per hour and so forth, I would not call any one of them an "index," not because they are not indexed to something, indeed they are, it's just that they are physical measurements and calling them indices would be besides the point. Now an index can be defined as anything that is not physically interpretable in the context in which it is used but provides an indication of some target physical quantity is called an index. For example from medicine, a rate-pressure product would be heart rate times systolic blood pressure. which is indexed to cardiac work load, but cardiac work would be more accurately defined as both pressure-volume work and blood volume ejected acceleration work. So rate-pressure product is what we would call a quick and dirty approximation, and is indexed to cardiac workload.
Generalizing the above, we can define anything that is not an exact physical quantity can be called an index. So body mass index (BMI) which has units that do not relate to anything to do with how fat people are is most definitely an index. One could make the claim that AIC is an index, although AIC proponents might balk at that. So most generally there are measurements, and things indexed to measurements that are not themselves physical in the context in which they are used.
This answers both of your questions. Maybe this will help: "measurements measure and indices indicate."