Your syslog should show something like....
Jun 4 21:00:20 pi4 weewx[10457] INFO user.belchertown: New forecast file downloaded to /home/pi/weewx-data/public_html/weewx_belchertown/json/forecast.json
And the timestamp of the first record of that forecast.json file should match that date....
{"timestamp": 1685937620, "current": [{"success": true, "error": null, "response": {"id": "KSEA", "dataSource": "METAR_NOAA", "loc": {"long": -122.31666666667, "lat": 47.45}, "place"
: {"name": "seattle/metro", "city": "seattle/metro",........
pi@pi4:/var/www/html/weewx/weewx_belchertown $ date -d @1685937620
Sun 04 Jun 2023 09:00:20 PM PDT