Yes, a single convex lens is a telescope, as your diagrams show, but only if the lens is held away from your eye, and the image in the lens appears inverted. Notice: the focal length of the lens must be significantly smaller than the length of your arm!! Find a lens with a fairly long fl, but not too long.
To produce telescopic magnification, move the lens away from your face until the image seen in the lens becomes extremely large and uselessly blurred ...then move the lens even farther. The image now should be upside-down, large and blurred. Next, cross your eyes, hard. While they're crossed, look at one of the pair of views, and find the lens. Everything in the lens will be clearer! Move the lens in and out until the image is sharp. It will have some low-power magnification.
By crossing your eyes, you also adjust them to focus on very nearby objects. The real-image produced by the convex lens is behaving as a nearby object. This real-image is close to your eyes, and hence larger. By crossing your eyes, your eyes remove the blur. They become the eyepiece of a Newtonian (inverting) telescope, and the image seen through the lens becomes sharp and clear.
If your eyes could focus on objects a few mm in front of your cornea, then even with no eyepiece used, any convex lens could behave as an inverting telescope of several-power magnification. (Also, you could use such an eye to examine insects in great detail without needing a hand lens!)
More: if you're an extreme myope, and can clearly see objects held very close to your eyes, then you can use a single convex lens as an inverting telescope, no need for crossed eyes, and no eyepiece. Your own eyes serve as the eyepiece.
And, if you're an extreme presbyope, far-sighted, then you can use a single convex lens as a Galilean or 'terrestrial' telescope. In that case look at a distant object, hold the lens close to your face, then move it away until the view becomes clear and sharp. The view will be magnified, but won't be upside-down as described earlier. Your eyes are acting as the concave eyepiece of a Galilean-type telescope.
Simplified explanation of telescopes: when viewing distant objects, the Objective lens produces a real image, a floating or "aerial image" positioned out at the focal plane of the lens. To produce large magnification, we place our eye very close to this aerial image, then add a "corrective lens" to our eye in the form of an eyepiece-lens to remove the blur. If the real-image had been placed just outside our eyeball, then the image will be inverted, the eyepiece lens must be be convex, and the resulting telescope type is the Newtonian. But if the real-image was placed inside our eyeball (closer than zero distance!) then a concave-type eyepiece is used, the image will be erect (not upside-down,) and the type of telescope formed is called Galilean. In general, the Newtonian type is superior because of field of view and abberation issues. So, most small telescopes and binoculars use this inverting type of 'scope, but then they add an "image erector" composed of a pair of 90deg prisms acting as four mirrors. Expensive binoculars may use a Newtonian with a Dove-prism or Roof-prism erector rather than two 90deg prisms. Very inexpensive binoculars or 'opera glasses' just use the inferior Galilean setup, with concave eyepiece and no prisms for erecting the inverted image.
The most inexpensive telescope of all uses a piece of aluminum foil with a tiny pinhole:
DIY telescope with foil-pinhole "eyepiece"