The Jewish definition comes from the Tanakh (The Jewish Bible). Messiah is found in Daniel 9:25 and 26 (haMoshiach). Moshiach is translated in various other places as "anointed" or "consecrated". It is used 39 times in the Tanakh. In Leviticus 4:3 "If the priest that is anointed do sin according to the sin of the people; then let him bring for his sin..." Since the priest sinned, he certainly isn't "The Messiah" according to the Christian view. To be defined as "The Messiah" it must be prefixed by the sign of direct object "ha" which is normally translated "the". The same is true for "haSatan". For satan to be "the Satan", it must be prefixed by "ha". Otherwise the term is understood to be any adversary.
Jews see HaMoshiach to be a man who is anointed, for the role of King of the World, just as the priest in Leviticus 4:3 was anointed for his role. Christians and Messianic Jews believe HaMoshiach is G-d.
The term "personal messiah" in all lower case letters indicates traditional Jewish views on the messiah. Using "Personal Messiah" with capitol letters indicates belief in Yeshua as Messiah.
Traditional Jews who believe in a personal messiah typically do not believe that messiah will be deity. They are looking for an above average human male to come and physically sit on the throne of his father David and rule from Israel. Many Orthodox Jews believe that messiah will have abilities beyond normal for man. He will fall somewhere between heavily anointed to G-d Himself.
Jews who have accepted the traditional view of Christianity's Messiah without giving up their traditions and refusing to adopt non-Jewish customs, are called "Messianic Jewish". Gentiles who accept the messiah are called "Messianic Believers". This term is generally applied to everyone who believes Jesus is Messiah, both Jew and Gentile. The term "Messianic Jewish" is reserved for Jewish believers in Yeshua.