Learn English – What’s the meaning of “by one count”

As mentioned by Maulik in his comments the word out in such contexts means somewhere away from a position.

However for this particular example it has a very specific meaning. In fact Out in "Out West" does not has its individual meaning.

Out west here is used as an Idiom that refers to the "Western part of the United States". You can find the reference here.

It usually refers to the states of California, Idaho, Nevada, Arizona etc.

They are normally equivalent.

On my count means I am going to count (down) and when I reach zero, you do what I told you to do.

On my mark means I am going to mark (indicate) the moment that I want you to do what I told you to do.

In both cases, the possessive is used to indicate that it is the speaker who will mark the start of the action.

That is not always the case; sometimes someone will say on three!, and then several people count together one, two, three - for instance when they are cooperating in a combined physical effort, the counting can help them coordinate their strength together.

On your marks refers to different marks: they are the markings on a track where the athletes should position themselves before a race. The_marks_ in on my mark is a marking of a point in time.