Does a UK residence card count as an Article 10 residence card?

The card you're asking about would allow your wife to travel to Iceland without a visa. She can also travel to Iceland by getting a visa. That is probably the way to go, as outlined below.

First, does your wife's current permit say that it is a "Residence card of a family member of an EEA national"? If so, it is an Article 10 (or Article 20) card. It probably does not say that, however, because family of British nationals do not usually fall under EU law in Britain, but under British law.

If her card doesn't have the magic text on it, then she should apply for a visa, for two reasons:

1. It's not clear whether the UK would grant her such a card, since she has another residence permit already. In any event, she is likely ineligible for such a card, because you are British.

2. Even if she were eligible for the card, the process for getting it is cumbersome and time consuming; it also carries a fee of £65. By contrast, Iceland will grant her a visa "free of charge as soon as possible and on the basis of an accelerated procedure."

She should therefore apply for a Schengen visa (see http://www.iceland.is/iceland-abroad/uk/visit-iceland/visit-iceland/). Because she is the family member of an EEA national (you), she will skip several questions on the application, and the application will be free of charge. She should get the visa in a couple of weeks.

Denmark handles Schengen visa applications from the UK on behalf of Iceland. They use VFS as a service provider. VFS services normally carry a fee. If the service cannot waive the fee, you can avoid it by applying directly to the embassy. It's not clear whether in such a case you should still apply with Denmark or Iceland, but I'm sure they can tell you.

The Danish site for EU/EEA family visas is available at http://storbritannien.um.dk/en/travel-and-residence/short-stay-visas/family-members-of-eu-or-eea-nationals/.

Your wife's case on that page is item 5:

5. Holders of Residence Permit on the basis of being married to/partner/child/parent of a UK national travelling with the UK national you are dependant on: visa required (free of charge)

In the comments, you wrote

I called the Icelandic embassy in London before; they told me they could not answer any visa related questions as this had been "outsourced" and gave me a telephone number which turned out to be VFS Global. The guy I spoke to at VFS Global did not seem to have a clue about this visa route!

I would try calling the Danish embassy.

Additionally, if I have to pay, I'd rather pay for an article 10 residence card than a VFS Global fee for a visa, as this will help us to enter other European countries in future.

As you note later, you are not exercising treaty rights in the UK because you are a British citizen. Your wife is therefore not eligible for an Article 10 card.

So I'm still wanting to know if a UK residence card counts as an article 10 residence card?

It does not. A residence card is an Article 10 (or Article 20) card if and only if it identifies the bearer as a family member of an EU/EEA citizen.

From what I understand, the residence permit is different from the residence card ...

Correct.

... and having the former does not prevent you applying for the latter.

That is probably incorrect. The two documents are issued under different legal frameworks.

A residence permit is issued under "national law" -- in your case, the immigration rules. These rules apply to everyone except for (non-UK) EEA nationals and their families.

A residence card is issued under Directive 2004/38/EC; in the United Kingdom it has been transposed into national law in the The Immigration (European Economic Area) Regulations. These regulations do not apply to UK nationals and their families except through the Surinder Singh exception (that is, a UK national returning to the UK after exercising treaty rights elsewhere).

Even if you could qualify for treatment under the EEA regulations, your wife would probably have to relinquish her current residence permit in order to get the residence card. This would obviously be a terrible mistake with Brexit looming, as you would probably have to reapply for a new residence permit at some point in the future. It seems much more uncertain and burdensome than simply applying for free visas every now and again.

PS: The page you quoted is for "Family Members of EU/EEA nationals exercising Treaty Rights in another EU/EEA country".

The page title is actually "Family Members of EU/EEA Nationals"; this is immediately followed by the lower-level heading you quote. I argue below that items 5 and 6 should be under a second lower-level heading that would make it clear that these items apply to people living in their own EU/EEA countries (or outside the EU/EEA).

Most of the material on the page does apply to family of those exercising treaty rights, but item 5, quoted above, applies to you. The mention of "UK national" in that item is, as you note, contrary to the phrase "exercising treaty rights in another country." Similarly, item 6 would apply to your wife if she were to travel without you.

The reason for this is precisely because, as you note

Articles 5 and 10 of the Free Movement Directive do not make any mention of whether the national must be exercising Treaty Rights or not!

Exactly, and that is why Denmark should give your wife a free visa to travel to Iceland.

The other important article for you is Article 6, which allows you and your wife not only to enter (Article 5) but also to remain for up to three months:

Article 6

Right of residence for up to three months

1.  Union citizens shall have the right of residence on the territory of another Member State for a period of up to three months without any conditions or any formalities other than the requirement to hold a valid identity card or passport.

2.  The provisions of paragraph 1 shall also apply to family members in possession of a valid passport who are not nationals of a Member State, accompanying or joining the Union citizen.

I skipped one sentence in your third comment:

I'm not an EEA national exercising Treaty Rights, because I'm a citizen of the country I live in, and so can live/work there due to my citizenship, not due to Treaty Rights... This is also explicitly stated on the page: "please note that UK nationals living and working in the UK do not exercise Treaty Rights".

That is precisely correct. Note that the quotation on the page is in the context of the Article 10 card; they're just trying to underscore the fact that your wife's permit is not an Article 10 card, and that she can't normally get one because you're a British citizen.

So, after they establish the difference between a card and a permit, they discuss the visa-free travel allowed by the card, and the next mention of the permit is items 5 and 6.

It would be less confusing if those items were under a different heading, "Family Members of EU/EEA nationals not exercising treaty rights because they live in their own EU/EEA country or outside the EU/EEA." Despite the fact that they are not, however, they do indeed apply to you and your wife.

Finally, I notice on re-reading that page that I missed the fact that it concerns travel "to Denmark or Iceland" so you should, without a doubt, follow the instructions on that page. It links to the regular "where to apply page," which in turn points to VFS. The VFS page says

There is a service charge of £ 25.69 (inclusive of VAT) applicable per visa application.

You should be able to avoid this by applying directly at the consulate. This is from the Handbook for the processing of visa applications, part III (Specific rules relating to applicants who are family members of EU citizens or Swiss citizens), section 3.2:

3.2. Service fee in case of outsourcing of the collection of applications

As family members should not pay any fee when submitting the application, they cannot be obliged to obtain an appointment via a premium call line or via an external provider whose services are charged to the applicant. Family members must be allowed to lodge their application directly at the consulate without any costs. However, if family members decide not to make use of their right to lodge their application directly at the consulate but to use the extra services, they should pay for these services.

If an appointment system is nevertheless in place, separate call lines (at ordinary local tariff) to the consulate should be put at the disposal of family members respecting comparable standards to those of "premium lines", i.e. the availability of such lines should be of standards comparable to those in place for other categories of applicants and an appointment must be allocated without delay.

Of course, it may be worth your while to pay the VFS fee if it is more convenient than going to a consulate or the embassy. The Danish embassy is in London. The Danish government lists nearly two dozen consulates and vice consulates in the UK. If a vice consulate can't help, I'd try the closest consulate if it's closer than the embassy.