Must one make up prayers performed with the intention of "perhaps I will catch [the congregational prayer], and if not, I will make it up," along with doubt about their timing, and does this intention fall under the category of deliberate action?

The prayer of someone who wakes up late for Fajr prayer is valid as long as they intended it for Fajr, even if they did not intend ada' (performance within time) or qada' (making up missed prayer), or if they intended one over the other. This is the preponderant view among scholars, due to the lack of a condition requiring the intention of ada' and qada'. Al-Nawawi mentioned its validity in "Al-Majmu'" and "Rawdat al-Talibin," as did Shaykh Khalil al-Maliki and Ibn Muflih al-Hanbali. Your prayers that you performed with the intention of qada' are valid, and you are not required to repeat them, even if they were before the time for ada' had expired. It is advisable to take measures to help oneself wake up for prayer.