Is what we did—passing the Miqat without consecrating ihram, then consecrating ihram from Medina after staying in Jeddah—considered a mistake similar to passing the Miqat with the intention of Hajj without consecrating ihram?

There is no blame upon you for what you did, because you did not intend 'Umrah with your trip to Jeddah. Therefore, it was not obligatory for you to enter into ihram from the miqat when you passed by it. Rather, you intended 'Umrah from your trip from Medina, so it is obligatory for you to enter into ihram from its miqat (Dhu al-Hulayfah). Sheikh Ibn 'Uthaymeen, may Allah have mercy on him, was asked about this and he permitted this action. Furthermore, prayer in the Sacred Mosque (Masjid al-Haram) is better than prayer in the Prophet's Mosque (Masjid an-Nabawi).