Well you've just got to ask: "What was Jesus trying to say in context?" That's the best way to determine the meaning of a particular word which has a broad range of meaning.
For example, the english word "bank" has a broad range of meaning (just like ginosko). It can mean "the place where I put my money", "the side of the river", or even "the way a basketball goes into the hoop". You've got to look at the context to discern what particular meaning is being used in any circumstance.
Same with Jesus here. His point is just as we've said - that he never had a saving relationship with these people, even though they believed they were saved. They did many mighty works in his name. The broad point is that "not everyone who says to me 'Lord, Lord' will see the kingdom of heaven." So we should be satisfied with the simple meaning that these people did not know Jesus in a saving way - nor he them.
But this should not be stretched to mean that he was not aware of their existence. This would be absurd, as he is omniscient God. And it should not be stretched to mean that he had no relationship with them at all. He is the king of the nations and so has an authoritative relationship with all people - rebels and saints alike.
