Yes, there is a difference between completed tests and tests taken. Some tests now give results in an hour though.
I just wanted to highlight the last bit, "don't visit your grandma for a few weeks". Nobody has had any exposure to this virus prior to december, it is completely new to everybody. You will only get immunity by getting it or getting a vaccine. Vaccines will take about 12 months (optimistically, there are about 20 different vaccine trials starting up now). So it might be to not visit your granda, or your grandchildren, until you're vaccinated or have tested positive, and that might take a lot longer than a few weeks.