Those words are not synonymous and the rabbis you are speaking about below, also state that.
And if you look at their videos they make very clear distinctions with what Chritianity believes a "messiah/deliverer" to be.
The Talmud was compoed after the supposed time of jesus, whoever he historically may have been. The information compiled in the Talmud also covers time frames after the time supposidly the historical jesus may have existed. Also, none of the views in the Talmud about the future Davidic king are halakha. Further, none of them beleived in jesus.
Actually, the Torah based Jewish view is that the NT does not record actual history. I.e. the jesus that the NT authors talk about didn't historically exist. Thus, there is no rejecting anyone. Simply put the NT authors were more than likely influenced by either an individual or individuals and when said person(s) did not accomplish what he claimed the early Jewish Christians had to make something up to cover for why things didn't go the way they expected. Then when they were starting to fade away historically a few of them took their beleifs to non-Jews in the Roman empire who bought into it.
No word games. Simply put, the national and cultural langauge of the Israeli/jewish people is Hebrew. Always has been. The language of the NT and the early Christians was Greek. Hebrew and Greek have vast differences in linquistic history and cultural nuance. Further, since most of Christian history is made of non-Jews who did not know Hebrew and many of them not known Greek. What you have is Christian ideas and concepts on a hill and Jewish Hebrew based Tanakh and concepts/ideas in a completely different country and the other side of thirteen rivers.
The good news is that Jews are not trying to convert Christians to Jewish ideas. We are not here to be the thought police on such topics.
Okay. I am going to let you in on a little secret. As far as most Jews are concerned, Jesus beleif can be whatever Christians want it to be.
Torath Mosheh Jews have a command from Hashem to stay out of said belief, no matter form it takes and what direction it goes in. Basically live and let live.