A death sentence isn't exactly a "suggestion", and by "he could have left" you mean "he could have escaped but chose not to". (I believe this is the premise of Crito).
Not to beat a dead horse... but he drank the poison. He wasn't killed, he took his own life. There was an alternative in which he lived, but which he refused.
But all this is getting off the point. Honorable suicide was a custom in more than ancient Japan.
If you read the accounts escape would have been quite easy. His followers urge Socrates to leave and agree to help him (he had no small amount of supporters). But he says no.
When you kill yourself, even if the government has told you that you have to, it's suicide.
The same argument you are making that it is not a suicide could apply to ancient Japan as well.
