The definition of a Public Place in law is "Any place to which at the material time, the public are permitted to have access, whether on payment or otherwise." In other words, if it's not secure private property,and anybody can enter or leave at will, it's a public place, even if you have to pay to enter.
The term "lawful authority" covers a wide range of circumstances, from armed soldiers to someone carrying out a contract for payment.
"Reasonable excuse" means that the act which allegedly contravenes the law is, in the cirumstances pertaining at the time, legal because the accused is carrying out a lawful activity, and that the alleged offence relates to some aspect of that activity which is essential to the pursuance of the activity.
In relation to knives, this means that if you are "camping" (a lawful activity) and intend to use a fire to cook on (also a lawful activity), then it is reasonable that you should carry a knife (or axe or machette) for the purpose of preparing wood for the fire, and a knife for the purpose of preparing the food to be cooked on the fire.
In the case of bushcraft, or any other outdoor activity, you would need to demonstrate to a court that you were camping, and using a fire, or were travelling to or from a place where you were planning to camp, or carry out a similar activity, and that the bladed tool was essential to the performance of the activity, to have a complete defence.
It is also an important point that you only have to prove your defence on the balance of probability, but the prosecution have to prove that your reason is false "beyond reasonable doubt". Imagine the look on the police officers face if, in the witness box, he is asked by your lawyer, "Tell me, Officer, what else was my client carrying at the time when you arrested my client?", and has to reply, "A tent, matches and a pan." He has just proved your defence for you!
Hope this helps.
Alan