That brings a third party into the situation.
No it doesn't. The right to choose is the right to choose. Anti-abortion conservatives "always demands everyone adhere to their beliefs and demands rather than choosing for themselves." Claiming it's different because it's a third party is nothing more than a rationalization for getting an exception to tell other people what they can and cannot do. Conservatives don't get to be the sole spokesperson for this third party.
Not the same thing as telling me I'm not allowed my right because you don't care for it or from the other direction me telling you do what you wish but I'm not foregoing my right.
Well, you're reformulated your statement into a different context now. But now that you have put it in the context of rights, it's exactly the same thing. If you look at the history of abortion you'll notice that it was an uncontested natural right for thousands of years and it has only been relatively recently where conservatives decided to take that right, along with the right to privacy away. Row v Wade restored both the right to privacy and the right to choose.
Women around the world have used abortion to control their reproduction at every point in history, and in every known society. In the US abortion was widely and freely commonplace until about 1880 when states started banning it. Anti-abortion legislation was part of a backlash championed by religious conservatives against the growing movements for suffrage and birth control in a blatant effort to control women and confine them to a traditional childbearing role. These laws were also a convenient way for the medical profession to tighten its control over women’s health care, as midwives who performed abortions were a threat to the male medical establishment. In addition, the late 1800s saw a declining birthrate among whites, the U.S. government and the eugenics movement were concerned about “race suicide” and wanted white U.S.-born women to reproduce.
So while you want to rationalize a plea for this third party, the entire history of anti-abortion legislation is all about power, control, and telling other people what they can and cannot do.
Conservatives aren't even pro-life, they're anti-abortion. They invented the pro-life nomenclature to demonize the opposition, because the opposite of pro-live is pro-death, and isn't that awesome. But they're really anti-abortion, and there's a difference. Pro-life means encouraging and celebrating life, but that's not what pro-lifers do. They're anti-abortion, which is power and control and telling other people what they can and cannot do (authoritarian). And once you realize that pro-lifers are simply anti-abortion, that they're pro-telling other people what to do, then there's no incongruity between "pro-life" and "pro-death penalty" because it's the same thing, power, control, and authoritarian arrogance. And it's why after failing to convince women they shouldn't have an abortion, that conservatives are trying to force others to live by their views by re-legislating authoritarian anti-abortion laws.
Yes, both the right and the left are in a never ending tug-of-war in trying to force their way on everyone. Neither side has a monopoly on it, and both sides are nearly always wrong. They only time either side is right is when you get about an 80% agreement all around.