The definition of "a public place" is interesting.
It means different things for different reasons. As has been mentioned, a machete in your front garden being used to chop back vegetation is no problem. However, waving it about and threatening passers by will see you on the wrong end of (at least) a Public Order Act offence even if the threats occur from well within the property's boundary. It has been ruled that the POA applies in such circumstances even though the offence is being committed whilst on private property.
For driving matters the issue is slightly different. Most of the Road Traffic Act applies to a "Road or other public place". Here, the ownership of the land in question is not material. The important issue is whether the public (in the form of other vehicles or pedestrians) have "unfettered access" to the area. This means that a supermarket car park, for instance, is almost invariably seen as a "public place" for these purposes even though the land may be owned by Tescos. However, an office car park to which only a few people have access and they gain entry by means of a controlled barrier would not be seen as public. The problem for the courts arises when there is, say, a car park that is "private" but which has no physical barrier, meaning anybody may drive into it, even though they are not authorised to do so.