Toe loop technique - are there two different options?

Beginner skater here who's begun working on her singles looking for some clarification ... I feel like I'm going crazy!

Are there two different mechanisms for jumping a toe loop?

My coach taught me right foot - inside 3 - toe pick into the jump, just like this.

When watching other skaters though, it looks like they enter with a left foot - outside 3 - foot down - toe pick like this or like this.

Is there something I'm missing? These seem like completely different jumps. Any clarification is so greatly appreciated!