Just to add to this, when a satellite goes into the Earth's shadow it disappears like turning off a light, and vice versa for a satellite going in the other direction coming out of the Earth's shadow.
I suspect this is it - you can't "see" where the shadow is but satellites go in all sorts of directions - and your eyes see a light moving and then another moving another direction and assume they're the same thing.
That was my first thought as well. It's not unusual to see several satellites overhead at the same time, and they can appear and disappear suddenly as they cross the earth's shadow boundary. Also they're quite faint and a brief discontinuity in their visibility is hard to spot reliably, or distinguish between blinking, glancing away for a moment, etc.
Have you ever noticed how sometimes when you look at your watch, the second hand seems to linger for slightly more than a second? This is due to saccadic masking. When your eyes move quickly, your brain cannot process the visual data. So it just backtracks and fills it in backwards with whatever it ends up seeing.
If the satellite passed into the shadow of the earth and dissappeared, your eyes could move to track another one without you seeing any perceptual gap at all.
Even if somehow that delta-v can be reached without burning so much fuel/energy that it makes any mid- to longterm mission fail, the forces on the satellite equipment which need to be withstood are insane. At orbital speeds it would just break the satellite/the craft unless there are some very special designs out there (?).
The typical pop-culture UFO shape, a saucer or a sphere, would be pretty good at distributing these forces across the "fuselage" I would guess, but internal systems are still affected. Anything with wings or solar panels would be toast, no?
Any abrupt change in heading is indeed pretty much impossible. A 90 degree plane change would require a staggering amount of delta-v.