Yes, in both cases that decision is made. But in euthanasia, a third party makes that decision without your informed consent. It could be someone that doesn't even know you or have any idea what your wishes might be. Assisted dying explicitly requires the individual to initiate and consent to the procedure. I disagree strongly that the difference is trivial. In euthanasia, the autonomy of an individual and her fundamental rights are being over-ridden. In assisted dying, the individual's autonomy and rights are being affirmed.

"The most dangerous step has already been taken by persuading people that it is acceptable to kill people whose lives not worth living."

The step you refer to was reached long before assisted dying was recognized as a human right. In many cases of terminal illness, doctors will at some point hasten a person's death by withholding treatments that could sustain life. That might be done without the patient's knowledge or consent- a form of euthanasia if you will.