Yes. That is exactly the rule. Weapons that are intended to injure but not kill are illegal, weapons intended to kill are ok. Injuring someone because you tried to kill them and missed is considered acceptable, because not everyone has perfect aim.
No, that is not correct.
The Hague and Geneva conventions forbid "To employ arms, projectiles, or material calculated to cause unnecessary suffering"
There are endless misconceptions regarding the Hague and Geneva conventions. Please read the actual text of the Conventions and updates. There are many surprises such as the circumstances that allow the execution of random civilians as punishment for the behavior of others.
Here is a link to the Hague Convention.
"Annex to the Convention: Regulations respecting the laws and customs of war on land - Section II : Hostilities - Chapter I : Means of injuring the enemy, sieges, and bombardments - Regulations: Art. 23."
The same web site includes the Geneva 1949 convention and subsequent updates (including the blinding laser weapons article)
Regarding other posts, nowhere are any type of bayonets mentioned. Almost no type of specific weapon is mentioned.
Many people confuse the US Army's rules of Engagement with the Geneva convention.
The ROE is much more specific and is the source of many of the things that people mistakenly believe are forbidden by the Geneva Convention.
FYI, the use of .50 cal machine gun against individuals is not forbidden in either place. No one knows where that story began.
If I am incorrect, please show exactly where it can be found in the Conventions.