A knife is a knife. Period.
I think you were asking about applications, but your wording seeks a factual definition. As a salesman, I have to get people to buy into ideas, behaviours, services and products. To do my job it is useful to undertsand the difference between features and benefits; simply put, features are and benefits do. Most people buy benefits to aid them in a task (application).
Its features (ie. what it is) are: a flat piece of material (usually a hard one) with some kind of thin edge; sometimes a point; and usually a comfortable place to hold it. The result of these features is that a knife can be used to concentrate force over a VERY small area.
If it has these features it is a knife, regardless of what it is used for (its application), sometimes they're just ornamental or educational.
These features usually mean that a force is concentrated over a very small area of another solid material (wood, paper, skin, muscle, hair, plastic, vegetation, metal, ice) and can cause the chemical bonds under that area to break, thus separating the matter in that area in a controlled way with maximum energy efficiency.
Thus the benefit of a knife (ie. what it does) is usually that it separates matter in a controlled and efficient way. It enables you to cut. For specific knives there will be other benefits, too, each related to specific features of that knife.
The application (ie. what we use them for) of a knife is limited only by your education and imagination, and sometimes by its specific features (grind, materials, shape, saw back, bell & whistles, handle shape, etc). Knives are usually used as tools to make objects or resources more useful, manageable, attractive or edible. Sometimes, making something more 'manageable', means cutting the weapon bearing, load bearing or life supporting structures of another animal, in order to disable them - in this case we may characterise the knife as a 'weapon', yet in this regard it is also still a tool.
So, what a thing 'is' can be looked at on several different levels:
1. Feature - it is a knife. Fact.
2. Benefit - it can cut. Subjective benefit (ie. a person may not agree that being able to cut something is beneficial to them). [So is a trainer a knife? What are the benefits of a trainer?]
3. Application - what it can be used for. Imagination and education will make this a really rich list, while ignorance or lack of experience will focus on the most easily understood/commonly communicated/emotive - ie. weapon. People who look at knives purely as weapons either have a fear of weapons/crime stemming from a lack of empowerment, or, driven by fear, have seek to empower themselves by delegating their fears to an object which they want to believe will give them the protection they crave. The former delegate their protection to authority, the latter to an object. Neither takes responsibility for their protection and neither get true protection. These people lack the vision to see the benefit in (2) and the imagination to understand the breadth of application in (3). This is not always their fault. In these cases the knife is a tool, whose application is weapon. The lid from a tuna can can be used to skin a rabbit, does that make it a knife? A knife can cause harm, does that make it a weapon? A car can cause harm, does that make it a weapon, also? Surely the answer to the last two questions must be consistent. If 'yes', does that make the tuna can a weapon?
Anyway, the short answer is 'tool'.