On the definition of "official"
The question posed in the title "What is an “official” supplement in the context of D&D 3.5?" does not have an answer, as far as I can tell. The word "official" is not defined in any D&D supplement to my knowledge.
But there are a few possible interpretations of the word, thanks to Wizards of the Coast's publishing scheme.
The Core Rulebooks
The core rulebooks (the Player's Handbook, Monster Manual and Dungeon Master's Guide) form the basis of D&D 3.5. This is defined on page 4 of the Dungeon Master's Guide, making it a self-defined system.
The above + content produced by Wizards of the Coast that explicitly declares itself as official
Some content produced by Wizards of the Coast, such as the Errata Files specifically call themselves official.
The above + other content produced by Wizards of the Coast
If we accept that Wizards of the Coast is an authoritative figure on D&D 3.5, its products, when marked with their own logo and published under their name, will be official content.
The above + licensed content by Wizards of the Coast
This is in accordance with a dictionary definition of the word "official", specifically "appointed or authorized to act in a designated capacity". Notably, this includes Dragon Magazine, when published by Paizo. Observe that at least the Dragon Magazines I have declare themselves as "100% official Dungeons & Dragons® content" on the cover, with no objections that I know of from Wizards of the Coast.
This, including the above categories, is what I consider a reasonable interpretation of the word "official".
The above + other content
There is a myriad of material produced using d20/OGL content based on the 3.5 system, such as the Rokugan campaign setting. These are published without Wizards of the Coast's involvement, and do not carry the "Dungeons and Dragons" logo. I can not find a way to define this as official.
On the categories of material you presented
The core rulesbooks
Must be official, as described above.
The main splatbooks and material from setting-specific sources
Are indistinguishable from a publishing point of view, assuming they come from Wizards of the Coast (note: some aren't affiliated with WotC at all, and would fall into "other content" described above). I can not find a way to call one official and not another - the licensing and legal language found on each one's first page is more or less identical.
Note that not all of these products are necessarily compatible with one another, or meant to be used together in a game. That's a separate issue that does not make any of them a less official D&D 3.5 product.
Material from Dragon Magazine
As I outlined above, Paizo had an official licence from WotC to produce Dragon Magazine. It self-declares as Official D&D Content, without complaint by the trademark holder. I find it reasonable to call it official. Whether it's a good idea to use it in a game is again a separate issue.
Web supplements are, of course, not conventionally published. I argue that they are still official D&D 3.5 material.
First, they are published on Wizard's of the Coast's website, which happens to carry a banner of "Dungeons & Dragons Roleplaying Game Official Home Page", which is a good indicator.
Second, and far more importantly, they are marked and published (as they are) with the Wizards of the Coast logo and under their name. Quoting the "More Divinity" web enhancement (the source of the Dweomerkeeper), it refers to itself as "This Wizards of the Coast game product". Despite not being printed, I find it hard to conclude that these supplements are not "official".
Note that online articles such as Skip Williams' "Rules of the Game" series do not fall under the second condition. Such articles are posted on the official website, but they are not signed and issued by WotC as an entity, just the author of each article.
Old material from 3.0
There is a certain amount of confusion regarding this. All I have been able to find on the interopability of 3.0 and 3.5 is the following clause from the 3.5 Dungeon Master's Guide
This is an upgrade of the d20 System, not a new edition
of the game. This revision is compatible with existing products, and these products can be used with the revision with only minor adjustments.
and the online update guide. I have not been able to find a source for the quotation that "anything not updated is automatically valid in 3.5" or similar. I believe it may be an overstated version of the actual DMG text.
The 3.0 products that are referred to are of course published by WotC, but their use in a 3.5 game seems contingent on "minor adjustment" by the DM.
The Living Greyhawk rules do not refer to such general terms as "official products". The last Living Greyhawk Sourcebook released for 3.5 contains an explicit list of sources that are "Open" for use in LG, see Appendix 4. It contains the whole Complete and Races series, and an assortment of other sources, including two Dragon Magazines.