The Trulia listing only gives you information about "nearby" schools, and doesn't specify the assigned school. The school closest to the house is not necessarily the assigned school; in fact, for your particular example, Lankenau High School is listed as the closest high school and it is not even located in the right school district. Public records for a property identify the assigned school district, though not necessarily the specific school within the district. You can get that information from the school district itself.
Gladywne is located in the Lower Merion School District, which has two public high schools: Harriton and Lower Merion High School. 1621 Briar Hill Rd is assigned to Harriton. Good luck and please let me know if I can be of any further help. I am a real estate agent and focus my practice on properties located within the Lower Merion School District.