When you are casting a spell, there are often choices that the caster makes, such as area(s), target(s), or damage type. If a spell takes longer than 1 action to cast, because it has a 1 minute or longer casting time or is being cast as a ritual, when are these choices decided? Do you make the decision when the spell is cast or when you complete the spell.

I typically don't need to pay much attention to the timing of spells with long casting time so I was surprised when I couldn't find anything on the timing.

With 1 action spells, bonus action spells, and reaction spells, this doesn't really matter.

I was partially wonder because of reactions like the Arcane Trickster's Spell Thief ability which requires the Arcane Trickster to be included in an AoE but that has its own timing requirement ("after a creature has cast a spell") and I can't think of another reaction with that kind of restriction that does not trigger after the spell is cast.

With no experience of an Arcane Trickster, can you give an exact example of an AOE spell that has to have the caster in it?  I am just not following this.

This question doesn't really matter, because with most (maybe all), long casting time spells, you'll have decided exactly what you want to do with the spell before you start casting it. No one's going to decide to resurrect the other guy halfway through the casting time of Resurrection.

Quote from SmiteMakesRight_3_5 >>

When you are casting a spell, there are often choices that the caster makes, such as area(s), target(s), or damage type. If a spell takes longer than 1 action to cast, because it has a 1 minute or longer casting time or is being cast as a ritual, when are these choices decided? Do you make the decision when the spell is cast or when you complete the spell.

 

A spell hasn't actually been "cast" until after the spellcasting "process" has fully completed.  Even then, there can be a distinction between the moment that the spell has been cast and the moment that the spell effect originates (the main example of this being when you Ready a spell).  

Anything that is related to spell targeting (both usages of "target" -- the selection of the point of origin as well as the selection of which creatures will be affected) is organized within a subsection of rules which falls under the umbrella of "Effects".  So, decisions such as positioning your AoE, targeting specific creatures and determining the damage type would be made at the moment that the spell effect originates.  For example, this allows you to Ready a spell while there are no valid targets within your view, but if such a target creature subsequently comes into view (because that creature moved OR because YOU moved), you can then target that creature as the spell effect originates.  So technically, by RAW, you can begin behind full cover and complete the casting via the Ready action so that you cannot be counterspelled (since no one can see you) and then you can move yourself out of cover and release the spell effect onto the target creature that you can now see.

So, to summarize:

A spellcaster decides to cast a spell and begins the process of spellcasting, which requires some time (the Magic action, for example) and some activity (V, S, M components, for example).  During this time the spell can be countered, but since the spell was never actually cast, the spell slot is NOT lost.

If the spellcaster was actually Readying a spell and was able to successfully fully cast it as part of that Readying process, then there is a period of time after the spell has been cast but before the spell effect originates.  This requires concentration.  If the spellcaster fails to respond to the Ready trigger within the allowed timeframe or if the spellcaster loses concentration during this time, then the spell effect never originates -- but the spell was cast during this process and therefore the spell slot IS lost.

In either case, once the spell effect successfully originates, decisions related to targeting are then made at that moment.

Quote from SmiteMakesRight_3_5 >>

When you are casting a spell, there are often choices that the caster makes, such as area(s), target(s), or damage type. If a spell takes longer than 1 action to cast, because it has a 1 minute or longer casting time or is being cast as a ritual, when are these choices decided? Do you make the decision when the spell is cast or when you complete the spell.

All those things are an effect of the spell and thus only takes place when the effect of the spell plays out, i.e when the casting of the spell is completed.

Quote from Thezzaruz >>

Quote from SmiteMakesRight_3_5 >>

When you are casting a spell, there are often choices that the caster makes, such as area(s), target(s), or damage type. If a spell takes longer than 1 action to cast, because it has a 1 minute or longer casting time or is being cast as a ritual, when are these choices decided? Do you make the decision when the spell is cast or when you complete the spell.

All those things are an effect of the spell and thus only takes place when the effect of the spell plays out, i.e when the casting of the spell is completed.

Making those decisions when the casting is complete also lines up with when a spell slot would get expended

Longer Casting Times

Certain spells—including a spell cast as a Ritual—require more time to cast: minutes or even hours. While you cast a spell with a casting time of 1 minute or more, you must take the Magic action on each of your turns, and you must maintain Concentration (see the rules glossary) while you do so. If your Concentration is broken, the spell fails, but you don’t expend a spell slot. To cast the spell again, you must start over.

The spell effect which include determining point of origin area or target occur after casting is completed.

Note that some spells (Planar Binding at least) say the target must be in range the entire casting time.  As it specifically has to say that, I would assume other spells would choose targets (and make other choices) when the spell is actually cast (ie, completed), unless it says otherwise.

There are some answers in the 2024 & 2014 SAC related to the question:

For readying a spell or another action, does the target have to be in range?
Your target must be within range when you take a readied action, not when you first take the Ready action to ready it.

If a spellcaster is affected by slow and takes two turns to finish casting a spell, what happens if their target has moved out of range or out of sight?
You choose the targets of a spell when you complete casting a spell, not when you start. (slow)

Quote from TarodNet >>

There are some answers in the 2024 & 2014 SAC related to the question:

For readying a spell or another action, does the target have to be in range?
Your target must be within range when you take a readied action, not when you first take the Ready action to ready it.

 

If a spellcaster is affected by slow and takes two turns to finish casting a spell, what happens if their target has moved out of range or out of sight?
You choose the targets of a spell when you complete casting a spell, not when you start. (slow)

Apart from this, is there any actual rule saying that you choose targets after the spell is cast, preferably, something updated for 2024? The 2014 SAC entry has a little less weight now that there is a 2024 version and it was not carried forward.

The 2024 Spell section on Effects say;

The effects of a spell are detailed after its duration entry. Those details present exactly what the spell does

A typical spell requires the caster to pick one or more targets to be affected by the spell’s magic.

So when casting for example Fire Bolt you;

You hurl a mote of fire at a creature or an object within range

After it take effect not during spellcasting; 

Well, since Casting Time and Components come before a spell's effect, and Targets fall under the Effect rules, the Targets are chosen when you complete casting a spell:

Effects

The effects of a spell are detailed after its duration entry. Those details present exactly what the spell does, which ignores mundane physical laws; any outcomes beyond those effects are under the DM’s purview. Whatever the effects, they typically deal with targets, saving throws, attack rolls, or all three, each of which is detailed below.

Targets

A typical spell requires the caster to pick one or more targets to be affected by the spell’s magic. A spell’s description says whether the spell targets creatures, objects, or something else. [...]

 
EDIT: ninja'd by Plaguescarred!