Hi, Graham! 真夏日 means the day when the highest temperature is higher than 30° , and 真冬日 means the day when the highest temperature is lower than 0°. 猛暑日 is difined as the day when the highest temperature is more than 35° , and 夏日 means the day when the highest is higher than 25°. I, as a native, don't think your way when I hear or read 真夏日. These words are just about the temperature. Therefore, 真夏日 can be in May, even though the middle of summer is July - the beginning of August. Hope this can help you understand!

I'm not a Japanese person, but I have confirmed that jisho is correct. I searched both words; according Weblio dictionary, 真夏日 is defined as "最高気温が摂氏30度以上の日." Basically, a day where the temperature exceeds 30 degrees Celsius. 真冬日 is also correct. 真 translates better to "true" than is does "middle."