I don't know what criteria non-Evangelical Christians employ to distinguish between a real and a fake Messiah, but afaik, all Christian denominations and sects believe in the 2nd coming of Christ.
The criteria are fundamentally different. In Revelation it is stated that war in the holy land will break out and that a third temple will be constructed. There however the antichrist will come, as well as the beast from the land and a beast from the sea. Together they will do many wonders and many people will believe them to be with/to be God and the Messiah. When in fact they, as usual, deceive the masses.
In secterian context it's rather important, because the Bible itself and Christendom overall makes a distinction between the Jews and the Gentiles who have found Salvation in Christ, the actual Messiah. Which boils more or less down to people who believe in Christ, his resurrection and on God to be the new Israelites/God's people under the covenant made with God through Christ. The Jews, as they are known today, are of the school of the pharisees, i.e. adhering to the Babylonian talmud, unrepentent dogma and in general opposition to the Gospel, as has been the case during the times of Christ on earth.
Most Christians reject the Ideology of evangelicals, because for one, you cannot force God's hand. It will play out when it will play out, and you are not even close to powerful enough to bring this about before the time has come, to which even Christ stated that not even he knows when this time will be, only the Father (God) does. To most, including to me as I consider myself a believing Christian, this is invoking a sense of itentions to bring about the antichrist. Which, as the false Messiah will be worshipped in the third temple. That jewish eshtalogy says that their Messiah will come with the third temple (and them having rejected the true Messiah before and after being raised from the dead), well you can imagine what I and others think about this. Ultimately, any Christian who is a Zionist or supportive of this is a strange case, because Zionist and jewish views are inherently incompatible with Christian views. There are more similarities between Christianity and Islam than between Judaism and any other religion.
Also Zionism is not completely detached from religion/judaism. It is a secular interpretation of judaism, as a substitute for regular ethno-nationalisn but it ultimately has it's entire foundation in the torah, tanakh and obviously the less than well looked upon talmud.