I'd get some 2X material of the same dimension as the joists and cut a couple of pieces to fit between joists 1 and 2. Attach them with joist hangers. Then attach the motor to one of the new crosspieces. If you get any movement of the joists when the door is being operated, then you should add additional bracing to the existing joists, perhaps running all the way from the front to the back of the garage.

Your rafter ties are meaty enough to hold some 2x framing you put over them. Just don't mount the garage door opener directly to whatever wood framing you install. It should still be attached via metal struts connected to that wood framing.

Alternative, a shaft drive opener. Much quieter and eliminates the hanging drive and motor mid-garage

So the OP already has the replacement/new motor and you want him to junk it and spend a fortune on a different style?

To follow up on this, I decided to add support between the joists with 2x and joist hangers. However, because the currently installed joists aren't completely level (slightly angled from top to bottom as a result of those extra braces added in and connect to the rafters, see pic). This causes a slight gap at the bottom between the new 2x and joists. Should I cut the new 2x at a similar angle ?

Don't know if your unit was sitting there like this for 13 days, but just lay the framing on top the rafter ties, since it appears your unit will be hugging the bottom of the rafter ties. Mount it to slotted steel angle in a strut system to hang your unit from. Thousands of videos on this, angle readily available at the big box.

These 2x8(?)@48"o.c. are rafter ties, where the connection to the rafters is a critical connection point to prevent your walls from bowing out. Those little purlins are not necessary and have done damage by twisting (apparently) the rafter tie away from the rafter heel.