The best way to decide whether to reduce a sauce or to add a thickening agent is to taste it.
If the flavour is as strong as you want it to be, then reduce it no further and add something to thicken it.
If the flavour is too weak, keep reducing it.
Other points to consider/caveats: reducing will increase salt concentration, so even if the flavour is too weak you might make a sauce inedibly salty if you reduce it too far some flavour compounds will get destroyed by too much boiling, so consider how the flavour might change as you reduce it
(e.g. lemon juice will lose some of its fresh, bright flavour, alcohol will lose some of its kick) various thickening agents will thicken the sauce in different ways.
it may be useful to familiarise yourself with them some thickening agents will have an effect on the flavour of the sauce (especially wheat flour, egg yolks, but also to a lesser extent cornflour)
cooking.stackexchange.com/users/50888/canardgras
You are on the right track.
Reducing is when you let the cooking liquids gently evaporate until the resulting sauce is concentrated to your desired taste and consistency.
Thickening is when some type of starch such as flour, cornstarch, arrowroot, etc.
is used to actually thicken the cooking liquids.
This is how you would make a traditional gravy or thicken the base for a stew.
Cooking liquids can be more than just water and can include things like meat juices, wine, broths, stocks, etc.
So, in order to thicken you need liquid, just not necessarily water.
Update -
The question was edited after I posted this answer.
There are a few different reasons to choose a reduction sauce or a thickened sauce.
A reduction sauce is typically considered to be more delicate and "cleaner" tasting.
(Not less flavorful.)
I typically drizzle a reduction sauce over meat and lighter vegetables such as asparagus.
A thickened gravy is typically heartier.
Most often you will see a thickened gravy used with heartier vegetables and sides such as potatoes or stuffing.
It is also used when cooking "smothered" dishes and stews.
In the end, just as with other things, it comes down to what you like.
In either case, there can be pitfalls.
With a reduction sauce you want to be sure not to reduce it too far as the flavor can become too strong.
With a thickened sauce, you want to be sure not to add too much starch as it can reduce the base flavor or drastically alter it.
cooking.stackexchange.com/users/26180/Cindy
It depends on the sauce.
A gravy you can thicken with flour.
I would never add flour to a tomato sauce.
Simmer also lets the flavors combine.
cooking.stackexchange.com/users/45636/paparazzo
It is all a matter of what tastes good to you and to those you want to please by feeding them.
There is no universal answer!
Some will taste better if reduced, some if thickened and some with a combination of reduction and thickening.
The only real answer is to experiment!
Try both ways or a combination with different recipes.
What is best for one dish, might not be right for others
cooking.stackexchange.com/users/66554/Cynetta
A lot of points have already been made, so I won't repeat them, but perhaps one thing is worth a thought.
A lot of Western sauces depend on thickening by creating emulsions of fat in water..
French-tradition reduction sauces are often thickened this way, with butter.
The mouth-feel is very different from thickening with starch - think of the difference in feel between mayonnaise and bechamel.
Also, emulsions can carry and intensify the flavor of fats - an olive oil which is very pleasant to dip bread in might be way too strong in a mayonnaise.
cooking.stackexchange.com/users/59328/Robin Betts
The approach I always adopted to thicken a sauce is to reduce it so the water can evaporate, leaving a sauce that is more concentrated. Earlier, I watched a video where the host was talking about thickening agents such as flour. Looking up on google, It seems that the thickening is a term commonly used when employing other substances to thicken the sauce rather than just letting the water evaporate. I wonder how to decide whether to reduce a sauce versus thickening it (or both)?
How to decide whether to reduce a sauce or thicken it?
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<sauce><thickening>
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cooking.stackexchange.com/users/68926/Amine Hajyoussef
A reduced sauce is often more delicate and 'cleaner' in taste, but the flavours and salt will be more concentrated and some flavour compounds can be destroyed by the extended boiling. Thickened sauces are heartier, but the flavours could be weakened or altered by the addition of starch. Some sauces work better with one or the other method, or a combination of the two, but you can taste and judge for yourself. Thickening and reducing are two ways of making sauce. Taste is the main factor when deciding on a method. In general reduced sauces are more delicate and thickened sauces heartier. In both methods, care needs to be taken to avoid spoiling the taste of the sauce.
Taste is the main factor in deciding whether to reduce or thicken a sauce. Reducing intensifies falvour but may make a sauce salty. Adding too much starch when thickening a sauce will alter the flavour. Reduced sauces are more delicate, thickened sauces heartier. Some sauces are more suitable to one method than the other, for example, never thicken tomato sauce.