What are the differences between stamped, pre-stamping, used, and CTO?

Philatelic terminology explained.

Stamped (1)
This means that a stamp bears a cancellation mark, rendering it no longer valid for postage. The cancellation mark may have been applied by the postal service when the stamp was used for postage, but it could also have occurred in another way.

CTO
This is an abbreviation for 'Cancelled To Order', or stamped on request. It concerns stamped postage stamps that have not ever been used to send a postal item. Usually, the stamps also still have gum, or were issued without gum. It is possible that the request was made by a collector. For example, by buying stamps at a post office and immediately asking to have them stamped to add to their 'cancelled' collection. It is also possible that stamps are affixed to something and cancelled without any mailing taking place (for example, on a card, envelope, leaflet, or whatever). A well-known example is an (unsent) FDC (First Day Cover).
However, the term CTO usually refers to stamps that are cancelled by postal administrations. For example, in the case of a new subscription for cancelled stamps. But it can also occur when stamps are released onto the market for collectors. Because the stamps are no longer valid for postage, they can often find their way to the collector at a (much) lower price.

Specimen

Another form in which stamps may no longer be valid for postage is a SPECIMEN overprint. This was usually used to send samples of new stamps to post offices, magazines, etc. Sometimes it was also used to offer (higher values) at a lower price to collectors (for example, with Australia).

Used
From the above, it is already clear that stamped does not always mean used. However, used is not always stamped either. The definition of a used stamp is that a postal item has been sent with it. In principle, the stamp then receives a mark that makes it unusable. Usually, this is indeed one or more cancellations. However, it can also be a stroke of a pen.
Sometimes a stamp has been used but not cancelled. Although officially not permitted, it could be reused after soaking. These stamps are often referred to as unused without gum, but are actually used without a cancellation. Especially with classic stamps without gum, it is often impossible to determine whether the gum disappeared through use or, for example, by soaking off a never-used stamp from an album. This, and the fact that stamps on postal items used to remain uncancelled much less frequently, certainly justifies the frequent use of the term 'unused without gum' for classic stamps.

False cancellation
Some stamps appear infrequently used for a specific reason. For example, because they were in circulation for a short time, were valid for a short period, or because the rate changed shortly after issuance. As such, unused stamps occur much more frequently. In these cases, we often see a false cancellation that was applied later, whether by the postal service or not. Usually, this is without a clearly legible cancellation date.

Pre-stamping

Some countries have sold pre-cancelled stamps to customers who had large batches of mail to send, meaning that not every item had to be cancelled individually. Sometimes the cancellation is already incorporated into the design (for example, in France, which issues different designs for this purpose); sometimes the cancellation is printed on an existing design. In these cases, whether a stamp is used or unused can only be seen on the reverse side. Without gum is used; with gum is unused.

Stamped (2)
A separately stamped stamp can therefore be:
1. Used
2. C.T.O.
3. Soaked off an unshipped item (for example, an FDC)

4. Falsely stamped

5. A pre-stamping (without gum)