There is no doubt in my mind that the statistic is accurate - and you are correct, the date was (or should have been) July 20th. I've always found it interesting that data records like this are linked to bulletins issued by a particular agency (eons ago it was the U.S. Weather Bureau) on a scheduled timeframe which has nothing to do with the actual meteorological event. The other side of that coin is that the intent of the bulletin has nothing to do with meteorological records.

There is also a bias toward the start of an event, but not the end of that event. Although not the case with Dolly, the use of a Special Tropical Disturbance Statement will often highlight the beginning of a Tropical Depression (advisories will be initiated at 11am), but no similar bulletin will be issued when a tropical cyclone becomes extratropical - it is simply noted on the final 6-hour advisory.

There is some interesting evidence though that supports Invest 94L as a Tropical Storm a full 12 hours before it officially became 'Dolly". The Invest 94L Model Input bulletin (issued by NHC) had the sustained winds at 35 knots at 20/06Z and 40 knots at 20/12Z. While it certainly takes more than wind speed alone to determine the formation of a Tropical Depression, it is also a no-brainer to realize that Invest 94L must have reached Tropical Depression status (meteorologically) at some point before it was recognized as a Tropical Storm. The lack of any official recognition (bulletin) of that change in the status of 94L doesn't change the recognition that 94L became a tropical cyclone at some point before Bertha became an extratropical one.

Regarding three simultaneous tropical cyclones in July, one other date came close - July 22, 1966. 'Celia' became extratropical on July 21st at 18Z and was still extratropical on July 22nd at 06Z. 'Ella' became a Tropical Depression on July 22nd at 12Z and 'Dorothy' became a Tropical Depression on July 22nd at 18Z - close, but not quite there.
Cheers,
ED