Who decides which of the "or" effects is applied?
When a card says "target [something] or [something else]", the "or" is part of the target description. While you are casting the spell, you choose the target, and the chosen target can be either one of those things. In the case of that specific wording, that means that you can choose either a creature or a player as the spell's target.

Note: The specific card you referenced, Searing Spear, has had errata as of the release of the set Dominaria. It now says


  Searing Spear deals 3 damage to any target.


And "any target" just means "target creature or player or planeswalker". Many other cards had similar changes.