Is an individual's claim to self-defense impacted by whether or not they used a tool/weapon?

To provide an initial answer, without getting to the specific of the 51 statutory standards for self-defense in U.S. states and the exact Canadian standard, as applied to your examples, I'll make some general observations:

• Self-defense is a justification for doing something that would otherwise be illegal (intentionally using force against someone else).

• Self-defense justifications include a sense of proportionality. It isn't proper to use more force than is reasonably necessary to achieve the defensive objective.

• There are basically two levels of self-defensive force: Deadly Force and Non-Deadly Force.

• Preventing some crimes and harms is statutorily justified to do using deadly force (e.g. to prevent a murder). To prevent other crimes and harms statutes only justify the use of non-deadly force (e.g. to prevent shoplifting). The exact list of crimes in each category varies somewhat. Historically, for example, there have been some U.S. states that have authorized the use of deadly force to prevent a rape that does not put the life of the victim at risk, and others that have not authorized the use of deadly force for that purpose (I don't know if that is still the case).

• Typically, law enforcement officers are also authorized to use deadly force in some circumstances in which a non-deputized civilian could not.

• As a practical matter, a use of force that causes death is presumptively considered to be deadly force, even if the means used are not inherently deadly in all circumstances. Meanwhile, a use of force involving the use of a "deadly weapon" such as a firearm, is presumptively considered to be deadly force, even if it doesn't actually kill someone.

• But both of these "presumptions" (and I am using that term loosely in this answer, rather than with its precise legal meaning), can be overcome with relevant evidence. Mildly shoving someone with hemophilia (or tossing a dish full of peanuts in the face of someone with a severe peanut allergy), without knowing that this person suffers from this condition, is not a use of deadly force within the meaning of laws justifying self-defense, even if it actually ends up causing their death. Similarly, proving that you intended to and did, shoot out the tires of someone's car, or shot their foot, instead of shooting to kill, would not always constitute a use of deadly force for purposes of statutes justifying the use of force in self-defense.

Getting to the specifics of the question, the majority rule would be that the use of deadly force is justified in most circumstances to prevent a home invasion burglar from harming you or other people in the residence, and to repel the home invasion burglar from the residence, although some jurisdictions would qualify this in one respect or another. The case that the use of deadly force is justified would be stronger if the home invasion burglar was armed than if he was not, and would be stronger if the homeowner was not physically competent enough to be confident of an ability to dispatch the invader in a non-deadly manner.

If deadly force was justified in that circumstance, it wouldn't really matter how you killed him, nor would it matter that you intended to kill him to defense your home and the people in your home.

If the law only authorized the use of non-deadly force in the circumstances, for example, because the burglar had seized an envelope full of cash and was fleeing the house, so you were really only using force to protect your property, rather than to protect your home or the safety of the people in it, at that point, then the analysis would get tougher.

If you intended to kill the fleeing thief in circumstances when only non-deadly force was authorized, the weapon you used wouldn't matter. You intended to use deadly force, the force you used caused the intended death, and you did those things even though the law didn't authorize you to do so in those circumstances.

If you didn't intend to kill the fleeing thief in circumstances when only non-deadly force was authorized, but you ended up killing him anyway (so that you didn't have a prohibited intent behind your actions), then the question would be whether your intent and belief that your actions would not kill him was reasonable under the circumstances.

If you caused his death with your bare hands, or with a less lethal weapon (in truth, there is no such thing as a non-lethal weapon), your belief that the thief wouldn't die from your use of force would be more likely to be seen as reasonable.

If you caused his death with a lethal weapon, your belief that the thief wouldn't die from your use of force would be less likely to be seen as reasonable.

If the jury (or a judge in bench trial) didn't believe you were reasonable in your use of force which you didn't intend to be deadly, then the jury (or judge as the case might be) would not allow a self-defense argument to prevent them from convicting you of some kind of homicide crime. So you would probably be convicted of some form of homicide (perhaps heat of passion manslaughter), although you might still not have the requisite intent for first degree murder in circumstances like that (so that your self-defense argument might end up providing you with an incomplete defense).

This provides a summary of the legal principles regarding self defense, stated for Canada but applicable in most important details in the US (where it is governed by state law). If someone breaks into your home, that does not give you an excuse to beat them up or shoot them (retributive force can only be exercised by the courts). If you say "Boo!" and they run away, you may not legally kill them or even slap them. If you can call the police and make them go away, you may not slap or kill them. It is possible that they will undertake an aggressive act against you, and assault you without provocation. You are allowed to use force to defend yourself in that case. An alternative is to retreat, if possible, but in Canada and most if not all states in the US you have no duty to retreat from your own home.

Even if a person acts aggressively toward you, that does not justify using force against the person. You may use force if you are under a reasonable apprehension of death or grievous bodily harm from the assault and have reasonable grounds to believe that force is necessary to prevent grievous bodily harm or death. The force that you use must be no more than is necessary. If it is plainly obvious that a punch in the snoot will put an end to the assault, you are not justified in blowing his head off with a shotgun because he broke into your home.

The law does not require you to be omniscient on the issue of whether the person is armed. If force is justified in self defense, the law does not discriminate in terms of using fists, tasers, or guns, at least to the extent that possession of tasers or guns is legal (in some states, felons have not been able to avail themselves of self defense when firearm were used in self defense).