What is cold-pressed honey?

This is most likely referring to honey processed without any heating, often simply called raw honey, or possibly cold-extracted or cold-processed. It likely has more flavor and color preserved than commercially-processed honey (which uses plenty of heat), but it may not be much different from gently heated honey. It's not actually cold (honey flows really slowly when cold), just not heated.

There are two primary ways honey would have been heated during processing. First and most common, most commercially sold honey has probably been heated substantially (perhaps up to 165F - 73,89ºC) to help melt crystals, allow impurities to separate, and to make sure everything's dead and safe. This definitely does affect the honey - it makes the flavor a bit more mild, and the color a bit lighter.

If you're getting honey from a very small producer (maybe someone with a hive in their yard), it's also possible that they actually heated it in order to separate the honey from the beeswax. The most effective way to get honey out of the comb is to remove the caps and use a centrifuge to spin the honey out of the frames, but for hobbyist beekeepers without access to an extractor who don't want to just crush and strain it, another method is to heat the comb, letting the melted wax (it melts around 145F - 62,78ºC) float the top for easy separation. This is pretty gentle heating, and doesn't have nearly as strong an effect as the commercial processes.

"Cold pressed honey" sounds like a nonsense to me and perhaps alludes to something like olive oil extraction.

I've kept bees for over 50 years and never heard the term before. The very little that I have read on the term is probably written by well-meaning guessers.

To be called honey (in the UK or the EU), it should not be heated for extended periods or elevated temperatures, nor should it be finely filtered. I.e. pollen grains should not be removed, nor should anything be added!

Honey is presented for consumption in two forms.

Cut comb, where a chunk of capped comb is cut from the frame. (Yes you can eat the wax)

Liquid honey (see extraction below). Note that all natural honey will eventually set. If it doesn't, it's not honey! To get it to revert to its liquid form, gently heat the jar in a pan of warm water.

Extraction of honey is either done in a centrifuge, where the cappings are first cut off (honey should not be extracted unless capped, this ensures a low water content and so prevents fermentation in the jar) and the frames are spun to cause the honey to fly out. It is then strained and "settled" to allow air-bubbles to escape and any detritus (bits of bee, wax, lumps of pollen etc. to sink to the bottom) the result is clear, pure, unadulterated honey. The honey frames may be warmed prior to spinning to about body temperature or slightly above to help the honey flow in cold conditions. Bees (like humans) maintain a body/hive temperature of 38C. so this is not elevated heating.

Some honey's such as that from Heather or Oil Seed Rape (or older honey) may set (crystallise) in the comb and so must be extracted by crushing in a honey press (is this where the idea of the association with olives comes from? Uninformed people trying to sound smart?). The honey that runs off is strained to remove larger wax partials and usually needs to be warmed after extraction to allow it to be bottled at a sensible speed (and to get an accurate amount into the jar).

I hope that this is helpful? In short, honey should not be adulterated, but with all expensive commodities, there is room for criminal activity. To be sure, buy your honey from a local bee keeper.

Edit: I should point out that heating honey to a temperature that will melt wax is not permitted (I have never heard a legitimate bee keeper use this method for honey meant for human consumption). It changes the chemical make-up of the honey. Commercially, honey may be flash heated to sterilise it and to remove tiny crystals that may be sites for crystallization, however this is not effective unless pollen is also removed (see above). Raw honey is perfectly safe to eat, however (due to a single unconfirmed case of infant botulism), we place a voluntary label to advise not to feed to infants less than 1 year on jars.