As you mentioned, there are both cards that refer to “chang(ing) (the/a/any)” target(s) of a spell or ability and cards that refer to “choos(ing) new targets” for them. Cards with any and all of these different wordings (save one) show up in tournament play in various places, so it's useful to know the differences between them. All of these different wordings work slightly differently from each other, and 114.7a-d explain how each such wording works.
Let's walk through the potential wordings one by one by looking at each subrule in turn:
114.7a If an effect allows a player to “change the target(s)” of a spell or ability, each target can be changed only to another legal target. If a target can’t be changed to another legal target, the original target is unchanged, even if the original target is itself illegal by then. If all the targets aren’t changed to other legal targets, none of them are changed.
Goblin Flectomancer is an example of a card that uses this wording; let's say you use it on something–what are your options? The first sentence here says that the change
must be to another legal target–you cannot use the Flectomancer to make
Murder target your opponent, or a card in their graveyard, or a creature with shroud, and you also cannot choose to have the target remain the same. The second sentence explains that if there is no legal option for changing a target, that target remains the same. The third sentence lays out what happens when you use
Goblin Flectomancer targeting a spell with more than one target: if you can't change
all of the spell's targets,
none of them are changed.
114.7b If an effect allows a player to “change a target” of a spell or ability, the process described in rule 114.7a is followed, except that only one of those targets may be changed (rather than all of them or none of them).
Spellskite is a card that uses this wording. As explained in the previous subrule, if you use Spellskite on a spell, you A)
must alter one of the targets if it's possible to do so, B) cannot make
Lava Spike target Spellskite or otherwise change targets in illegal ways, and C) if you cannot change one of the targets to Spellskite, it remains unchanged. The only difference is what happens if you target a spell or ability with multiple targets: only one target may be changed, because that's what the card says: change
a (ie, one) target. This differs from 7a because when you change “the” targets of a spell, you must change all the targets, not just one, and if you can't change them all, none change.
114.7c If an effect allows a player to “change any targets” of a spell or ability, the process described in rule 114.7a is followed, except that any number of those targets may be changed (rather than all of them or none of them).
Sideswipe is the only card that uses this wording. (And therefore, as you might imagine, it doesn't really show up in tournament play.) Again, when using Sideswipe, you cannot change targets in illegal ways, and if you cannot change a target, it remains unchanged. But here, you can change “any number” of the targets, and “any number” includes 0, so you can choose to change none of the targets, some of them, or all of them, at your discretion. In 7a, you are forced to change all of the targets, and if it isn't possible, none of them change.
114.7d If an effect allows a player to “choose new targets” for a spell or ability, the player may leave any number of the targets unchanged, even if those targets would be illegal. If the player chooses to change some or all of the targets, the new targets must be legal and must not cause any unchanged targets to become illegal.
Redirect is a card that uses this wording, but it's more commonly used as a rider when copying existing spells on the stack, as
Twincast does. Here, you have total freedom to change things–or not–as you will, as long as any changes you make are legal and don't cause any unchanged targets to become illegal. (So if you use
Redirect on
Searing Blaze, you cannot change the creature target and leave the player target alone if that would mean the player is now an illegal target.) Change nothing, change one target, change them all–it's all up to you.
TL;DR: When something says “change” targets, they
must end up different if possible. When it says “choose”, they do not. When something says to change “the” target(s), it means all of them, and if you can't change them all, nothing happens. Otherwise it means however many targets it says it does.
Hopefully this was helpful. :)