That’s not how it works. You’re correct when you say that from your point of view it’s Earth’s clock going half speed and from Earth’s point of view it’s your clock going half speed while you’re traveling away from Earth (or Earth is traveling away from you, both are equally valid), but that’s only true as long as the distance between you and Earth continues to increase at 86% of the speed of light. As you decelerate at your destination your reference frame continuously changes until you’re back in the same frame as Earth (or nearly so, we can assume the two stars aren’t exactly maintaining their relative positions). While you’re decelerating, from your perspective Earth’s clock speeds up and goes faster than yours, how much is determined by your rate of change in relative velocity. Earth’s reference frame isn’t changing (ignoring movement around the sun, galactic center, the great attractor, etc.), so the Earth’s perspective on your clock doesn’t change, the Earth sees your clock gradually speed up as you “slow down” until it’s going the same rate, but never faster. So once you’re back in the Earth’s reference frame both you and the Earth will agree that your clock advanced 5 years while Earth’s clock (and your destination’s clock, adjusted for any relative movement between it and Earth) advanced 10 years. This assumes a constant 86% light speed and ignores the time accelerating at departure and arrival so let’s assume very fast acceleration so it doesn’t change more than a couple days.
Edit: this is all completely ignoring gravity based time dilation from the spaceship climbing out of Sol’s well and going down the destination’s well and only considers velocity based time dilation. It would be more correct if you only considered two spaceships in a void where one accelerates to relativistic speeds and then accelerates back into the reference frame of the other.
That’s not how it works. You’re correct when you say that from your point of view it’s Earth’s clock going half speed and from Earth’s point of view it’s your clock going half speed while you’re traveling away from Earth (or Earth is traveling away from you, both are equally valid), but that’s only true as long as the distance between you and Earth continues to increase at 86% of the speed of light. As you decelerate at your destination your reference frame continuously changes until you’re back in the same frame as Earth (or nearly so, we can assume the two stars aren’t exactly maintaining their relative positions). While you’re decelerating, from your perspective Earth’s clock speeds up and goes faster than yours, how much is determined by your rate of change in relative velocity. Earth’s reference frame isn’t changing (ignoring movement around the sun, galactic center, the great attractor, etc.), so the Earth’s perspective on your clock doesn’t change, the Earth sees your clock gradually speed up as you “slow down” until it’s going the same rate, but never faster. So once you’re back in the Earth’s reference frame both you and the Earth will agree that your clock advanced 5 years while Earth’s clock (and your destination’s clock, adjusted for any relative movement between it and Earth) advanced 10 years. This assumes a constant 86% light speed and ignores the time accelerating at departure and arrival so let’s assume very fast acceleration so it doesn’t change more than a couple days.
Edit: this is all completely ignoring gravity based time dilation from the spaceship climbing out of Sol’s well and going down the destination’s well and only considers velocity based time dilation. It would be more correct if you only considered two spaceships in a void where one accelerates to relativistic speeds and then accelerates back into the reference frame of the other.