Understand “slug” as part of a web URL

As has been mentioned in the comments, although you have gathered together quite a few definitions and examples of usage, a common thread runs through them all in terms of what 'figure out' means.

In a broad sense of the word, to figure out something means to understand something clearly, however, what differentiates it from similar expressions such as to realise something or to find out something is the process involved in coming to that conclusion.

When we realise something or find out something, we are able to comprehend something that we couldn't comprehend before, but that may not have been our intention. Realisations can often happen by chance - something may spark a connection, and you then see it in a new light.

For example:

I just realised that there's an arrow hidden between the E and the x in the 'FedEx' logo!

Similarly with 'finding out something' - we may have come by the information passively.

I just found out that my girlfriend was cheating on me.

Figuring something out, on the other hand, implies a conscious effort to study a matter or situation and actively search for the answer or understanding of it. Rather than by chance or happenstance, it is normally done with a specific goal in mind.

For example:

• We need to figure out what went wrong.
• It took them ages to figure out how to assemble the chest of drawers.
• When you've figured the answer out, put your hand up.
• They spent 20 minutes figuring out how to open the box.

There is a broad range of examples above, but all of these exhibit intent to come to a conclusion.

In a scientific context, accuracy and precision are different things, and some of this difference carries over to these words' non-scientific common usage.

An accurate scientific measurement is one that is very close to the real value. A precise measurement is one that is repeatable with very small variations, whether it's close to the real value or not. An exact scientific value is one that has absolutely zero error: maybe it's a purely defined quantity like the number of centimeters in a meter.

In common usage, accurate describes being close to the correct value. Precise, on the other hand, emphasizes the small margin of error or paying attention to the smallest details. Exact means zero deviation from the correct value.

You should also know that accurate and precise can be applied to either a number or the person generating the number. Exact, on the other hand, is seldom used to describe a person and only describes the number.

Describing a number, say the number of people attending a concert:

I need an accurate count of the people at the concert. [says that you need a number that's close to the truth, but maybe 3,000 is a good enough answer even if it's not exactly correct]

I need a precise count of the people at the concert. [says that you need a number with a small margin of error, something like 2,945 give or take a few]

I need an exact count of the people at the concert. [says that you want a count of every single person with zero error]

Describing an accountant:

She is a very accurate accountant; she has not made a mistake yet. [close to the truth]

She is a very precise accountant; she tracks every cost down to the tenth of a cent. [very small margin of error]

You cannot say "she is a very exact accountant." Exact does not describe people.

There is another adjective, exacting, that is applied to people, but it means "having very high expectations [of others]."

She is a very exacting accountant; she makes me submit receipts for every transaction, no matter how small. [very high expectations]

Looking at your sample sentences:

1. Could be either "precise" (emphasizing "down to the penny") or "exact" (emphasizing "this is the correct value").
2. "Accurate" would be best. You might think "exact" (meaning zero error), but you don't use "exact" to describe people.
3. All three are possible here. I'd probably use "accurate".
4. I'd use "precision" here, in the sense of "attention to the tiniest detail."

Added examples:

1. Either "precise" (meaning "down to the minute or second") or "accurate" (meaning "correct"). "Exact" is fine but sounds less natural.
2. This actually brings up a different shade of meaning. I talked about these three adjectives as modifying a measurement or guess. Here you're using them to modify a real thing that that the speaker is guessing at. In that case, you cannot use the word accurate. You can use either exact (meaning absolutely correct) or precise (meaning known in detail).
3. Same as 2.
4. Same as 2.
5. All three options are possible. "Exact" and "precise" are pretty close synonyms, meaning "identical meanings in both languages." Using "accurate" is a little looser, meaning that there's no word in the other language that comes close to the meaning in English.