Yes, you are correct and look at this article in the New York Times saying we in the USA use a more strict version of the test which causes a much higher positive rate.
"The PCR test amplifies genetic matter from the virus in cycles; the fewer cycles required, the greater the amount of virus, or viral load, in the sample. The greater the viral load, the more likely the patient is to be contagious.
This number of amplification cycles needed to find the virus, called the cycle threshold, is never included in the results sent to doctors and coronavirus patients, although it could tell them how infectious the patients are.
In three sets of testing data that include cycle thresholds, compiled by officials in Massachusetts, New York and Nevada, up to 90 percent of people testing positive carried barely any virus, a review by The Times found.
On Thursday, the United States recorded 45,604 new coronavirus cases, according to a database maintained by The Times. If the rates of contagiousness in Massachusetts and New York were to apply nationwide, then perhaps only 4,500 of those people may actually need to isolate and submit to contact tracing."
