It depends on time and place. In Israel and Western Europe I will take the "democracy" side of the argument every time (which, in Israel, is paradoxically the side that isn't talking about democracy all the time because the definition is so distorted), but in places such as Turkey and Egypt I'll gladly go with the "anti-democratic" alternative if necessary. Needless to say I supported the failed coup.
What is the democracy side of the argument in Israel and the west and is it uniform across them all?
Not "the same", but similar for sure, I think. Essentially, the question here is whether one adheres to a rather electoral conceptualization of democracy or rather chooses to prioritize other concerns, which may fall into the category of secularism (Turkey) but can also be more liberal conceptualizations of democracy (Western Europe, Israel). It is a contrast between democracy in its crudest form versus "freedom"/secularism/minority rights, and while the exact situation differs, the contrast seems relevant in many cases.
In Israel, the right tends to prioritize Knesset decisions over more liberal-democratic inclined decisions or advise, such as by the Supreme Court or NGOs. By contrast, the left seeks to diminish the importance of Knesset decisions by involving many other actors (which is what they call democracy, and while I, as a political scientist, obviously see the theoretical case for doing so, I consider it a bit of a distortion because it doesn't have much to do with the "demos"). This is similar to what we see in Western Europe: "populists" and eurosceptics want the decision-making process to mainly take place in parliament and support referendums, thereby adhering to a very electoral definition of democracy, whereas most established, mainstream political forces think it is good to involve many other actors and make sure not much power is in the hands of parliament, which they see as important in guarding liberal democracy. I would say the contrast is not the same and the actors are not the same, but there are definitely important similarities.
In both contexts I support the ones who adhere to a more electoral conceptualization of democracy, because I believe it is important that large majorities agree with policies and there should be a very direct link between the majority's vote and the policies that are eventually implemented, more so than is now the case.