Can you use your own movement separately from your mount's movement?

Question

SevenSidedDie and myself came up with a quandary concerning Mounted Combat, or any mount at all, actually. When moving while riding, a mount that has been trained to take a rider or is otherwise pacified into accepting a rider one of two things may happen:

1. The Mount acts on your initiative count and can only take 3 actions: dash, disengage, and dodge. It can also be commanded to move (for free of course since movement is just something that happens and takes no sort of action to complete).

2. The Mount may act on its own separate initiative count, but it acts of its own accord and may act unpredictably, moving where it wishes and attacking targets if it wishes (which it can't do if you chose to command the mount instead).

That's straight from the rules and isn't controversial. The issue is, how does movement work in each scenario?

Scenario 1: Does the mount, since it's moving on your same initiative count and moving only because you are controlling where and how it moves, move in place of the characters movement? Or can the mount move its full speed, the character may dismount, and then move its full speed as well?

Scenario 2: Does the mount, since it's moving of its own accord and on a separate initiative than the character, move unpredictably of its own accord, and then you may move as well once your initiative count arrives?

Answer 1

The mount and the rider each have their own full turn with movement and action. When a player mounts an animal and decides to control it:

The initiative of a controlled mount changes to match yours when you mount it.

The mount's position in the turn order changes but it keeps its own turn on which:

It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

Therefore the mount can move its full speed and finish its turn. After that the character takes his own turn on which it can dismount using half of the speed and move for the remaining half. What the character cannot do though is to mount, ride, and dismount as it would require two character's turns.

There is one thing that requires clarification though. If I understand correctly, ties in initiative are resolved once and for all, which means that after mounting the player has to decide whether it is the mount or the character who acts first during the round.

The only thing I see that may hurt my reasoning is this sentence (emphasis is mine)

A controlled mount can move and act even on the turn that you mount it.

It makes things arguable, but for me everything else is solid enough to consider that there is a mistake and it should be “round” instead of “turn” in this sentence.