V起來2 marks the same onset-and-continue but keeps 起來 fused at the tail (笑起來 = burst out laughing). V起N來 has an object to place, so 起 and 來 split apart and the noun threads between them (下起雨來 = the rain sets in). The deciding feature is whether there is an object to split around.
Directional 起來 lifts the action up or off the ground (站起來 stand up, 拿起來 pick up). V起N來 carries no upward motion; it marks taking up and running with an activity. Ask whether anything physically rises; if not, it is the launch reading.
了2 reports that a new state now holds (下雨了 = it is raining now). V起N來 films the action at the instant it kicks off and keeps going. One states the state, the other films its start.
✗ 他當老師起來了 → ✓ 他當起老師來了 (the object sits between 起 and 來, never after 起來)
✗ 他竟然當起來老師了 → ✓ 他竟然當起老師來了 (the noun splits 起 and 來; they do not stay fused)
✗ 唱起來歌 → ✓ 唱起歌來 (verb, 起, object, then 來)
English take up teaching or start to teach puts one phrase ahead of the object, so learners leave 起來 fused and tack the noun on after it, instead of threading the noun through the middle.