Is the variable at ratio or interval level if it can take any value between 1 and 100 but not 0? I cannot determine using the definitions of ratio and interval variables.
This doesn't depend on the possible values but on interpretation and what is measured. If it's counts of something, it's ratio, as if one value is twice as high as another, it really means "twice as much" in a properly interpretable way (actually proper counts are absolute scales, which means that they fulfill the conditions for all lower scale types).
Many sources say that a "ratio scale" requires an "absolute zero point". Note that according to the "definition" that I have given, the zero has a special role, because one can't divide by zero. In fact "interpretable ratios" imply (at least in all practical situations, if not strictly mathematically) that the measurements to be compared all have the same sign (normally they are all positive). Zero is then a borderline value that is treated differently from all the other values, as it is not involved in any meaningful ratios. Usually there is a specific interpretation for this, with zero meaning the absence of anything to be measured (zero counts, zero length, zero weight etc.).
On the other hand, if it is for example a rating scale without properly defined meaning of the numerical values (like for example used by a film critic to rate films), it is not even interval (which requires differences to be properly interpreted, meaning that in a well defined sense the difference between 94 and 96 has to be the same as between 12 and 14, say) but only ordinal (all we can say is 96 is better than 94 and 14 better than 12, but there is no quantitative meaning to "how much better").