If I don't declare a not-yet-used engagement ring at airport security, am I smuggling jewelry?

You are only allowed to bring into a country x amount of good(s) (this varies per country). For example:

The UK law for arrivals from non EU countries states:

You can bring in other goods worth up to £390 without having to pay tax and/or duty.

Bringing in items worth more than this amount is liable to tax, again e.g.:

...(if) you go over your allowance you'll have to pay any Customs Duty, Excise Duty and/or Import VAT due (VAT will also be due on any duties that may be charged).

If you fail to declare any item over the threshold (i.e. you go through the green lane not the red lane). You will not have paid the relevant tax on said item and are therefore breaking the law.

Definition of smuggling is:

Smuggling is the illegal transportation of objects .... across an international border, in violation of applicable laws or other regulations.

So in the circumstances above, yes you are smuggling.

This is regardless of where on your person the item is. If you're wearing it you're still planning on bringing an amount of goods into the country. How does customs know that you don't plan to take off that expensive watch and sell it later?

You can (as stated in the related question Item held by customs until I pay tax; what are my options?) claim the tax back when you leave the country in question, providing you still have the item. This is what you should do by law.

I checked this out for Canada, and the rules seem to be deliberately vague.

As a visitor, you can bring certain goods into Canada for your own use as “personal baggage”. Personal baggage includes clothing, camping and sports equipment, cameras and personal computers. [...] You must declare all goods when you arrive at the first CBSA port of entry. [...] The border services officer may ask you to leave a security deposit for your goods, which will be refunded to you when you export the goods from Canada.

There is no mention of a value of personal belongings below which declaration is unnecessary. On the other hand, the Canada Landing Card does not ask for the value of your personal belongings, and there certainly isn't a long lineup of visitors to Canada volunteering that information at Immigration.

All of that assumes that the goods will not be left in Canada. It's phrased like that, so it doesn't matter that technically someone else is taking the ring out of Canada from the person who brought it in.

I would estimate that, in Canada, the authorities would have to demonstrate that you intended to leave the ring in Canada to successfully prosecute you for smuggling. If you wanted to be safe, or your ring was very expensive, you could declare it on entry. As long as you take it out of the country you won't pay duty.

If I buy a ring in the US, put it in my bag and do not declare the ring I am smuggling jewelry? How is that different from buying the ring, wearing it round my neck on a chain (or on my own finger for that matter) and not declaring it? Am I supposed to declare my own jewelry that I already paid for (including taxes upon purchase)?

First off it's not airport security you should be concerned about. It's customs at the destination country.

Whether you need to declare it to customs depends very much on where you are going, where you live and what you intend to do with the ring.

As a visitor most countries will let you take in reasonable personal belongings on a temporary basis. If it's extremly valuable they may require temporary import paperwork and a deposit but most of the time that is more trouble than it is worth.

You can normally bring back stuff that you bought at home and took abroad.

On the other hand if you are returning home with a ring you bought aboroad then unless it was very cheap you most likely need to declare it. Your country is unlikely to care that you already paid foreign tax on it.

And if you are taking a valuable item to a foreign country and don't intend to bring it back out then again you likely need to declare it.

If you are immigrating (changing your country of residence) there are often special rules that let you bring in more than the normal allowances without paying duty/tax but taking advantage of these rules may require specific paperwork.

Customs officers don't know for sure why someone is bringing an item into the country so they have to make educated guesses. Whether an item is in it's original box is one of the things they are likely to consider. In general when it comes to customs duty the burden of proof is on the traveller.