'19 Kids & Counting' Baby -- Laughing & Playing After Holy Land Health Scare

She had reportedly fallen gravely ill during a trip to Israel with her family last week -- but baby Josie from TLC's "19 Kids and Counting" made a speedy recovery ... and TMZ has obtained video proof.

Reports are circulating ... Josie was rushed to a medical facility after her "oxygen levels had spiraled down" -- but sources close to the family tell TMZ, the reports were blown way out of proportion.

We're told 21-month-old Josie was merely experiencing some cold symptoms -- and was taken to a doctor's office to get them checked out.