Perhaps the manufacturers forgot to ask your permission to allow them to use such terminology. I know exactly what it is to both a springer and a PCP and I said so in my first response.
From Air Arms web pages:
The Air Arms S5 Sporter series takes the proven 400 series format and gives it a new level of sophistication. The rifles smooth, virtually effortless, sidelever action cycles each pellet with a flex of the wrist.
From Weihrauch web pages:
Weihrauch HW100 and HW100S purposely designed Biathlon Sidelever
From FX Airguns web pages:
Right hand positioned sidelever with an ambidextrous mount to be able to relocate sidelever on left side for lefthanded shooters (depending on stock).
etc, etc, etc.
I'm thinking you understand the differences but chose nevertheless to stick to your questionable position.
For the unfamiliar reader, a side lever is indeed used to load the pellets themselves in the PCP rifles referred to above. In general airgun terminology, however, it shouldn't be conflated to include PCP repeaters. These are pre-charged pneumatics that use high pressure compressed air to power the airgun. Most use a lever of some kind to load the pellet into a chamber. These air rifles are referred to as PCPs or PCP repeaters.
In generally understood airgun terminology, side levers are a different kind of air gun. The term side lever refers to the cocking lever of a spring piston air rifle (and some air pistols) and it's located on the side of the action. The term distinguishes it from break barrel cocking and underlever cocking. (If in doubt check it out on a reliable website that has airgun terminology.)
If airgunners were asked to identify side lever models, they wouldn't be referring to PCP models. They would refer to springers such as the Diana 48 and Diana 52 and the FWB 300.
To illustrate, below is the venerable side lever Diana 48 (left) and FWB 300 (right). For comparison, an example of an underlever air rifle is shown below

The HW 97, a popular underlever, is shown below for comparison. (The lever that's literally under the barrel is the lever that cocks the springer.) The Air Arms TX 200 is another well known underlever.

The side lever loading system on the HW 100 below. This rifle and others like it are known as PCPs, not side levers.
