This is a Good Reason to have Well-Defined Terms that are Used Consistently
Although it is a fact that Michigan and WI do not have a formal Reciprocity agreement, WI does recognize a Michigan permit and MI recognizes all resident permits/licenses...:
http://www.doj.state.wi.us/dles/cib/conceal-carry/reciprocity
I think that "reciprocity" implies a statutory provision of two-way recognition which may or may not require a formal agreement.
For one-way provisions - "recognition" or "mutal recognition" are good terms. Examples:
Michigan recognizes Wisconsin (in fact all states' licenses for residents of those states), whether or not the other state recognizes Michigan.
Wisconsin recognizes Virginia non-Resident.
Texas and Louisiana have reciprocity (by administrative agreement authorized by statute).
Virginia recognizes Washington (but not vice-versa).
By operation of statute, Florida will recognize any state (resident only) that recognizes Florida.
Many states that do not require reciprocity agreements execute them to meet the requirements of the other state.
Restrictions on the type(s) of weapon(s) carried are set by the law of the state in which they are carried. However, they may also be limited by the terms of the license/permit on which the weapon is carried. If the state of carrying specifically states that all of the conditions of the license (which are imposed by the issuing state) apply, then the carrier must comply with the laws of both states. You can also make a argument that the license is only deemed valid by the issuing state when its own conditions are complied with. Thus if State A says that its licenses is valid only for handguns (invalid for say, a knife). Then you cannot carry a knife in State B which recognizes State A (because you do not have a valid license for State B to recognize), even if State B's own license covers knives. I do not know of a situation like this, but people should be careful of such traps.