Is there any single word in English to represent domain of "logic and science" together?

I would call the domain of science and logic rationality or reason. From Lexico:

rationality: The quality of being based on or in accordance with reason or logic.

reason: The power of the mind to think, understand, and form judgments by a process of logic.

Your friends could lead the "rationality movement", the "reason movement", the "let's be reasonable movement", or maybe the "Can we talk?" movement.

Another possibility is enlightenment. From Lexido:

enlightenment: The action of enlightening or the state of being enlightened.

the Enlightenment: A European intellectual movement of the late 17th and 18th centuries emphasizing reason and individualism rather than tradition.

Yes, the Enlightenment has already been used, but your friends could call their movement The New Enlightenment.

The proper term to use is Evidence Based. Join the Evidence Based movement!

The practice of Logic is based on known, largely agreed upon mathematical rules. These go from the syntactic to the semantic but I'm sure there is a modern host of variations and embattled subtleties. The distinction it enjoys is that there are rules.

Science is a bit more subjective, deal with it. The practice is to offer explanations of observed data. These are couched within the context of an hypothesis and a conclusion or results based on how well the hypothesis fits the observed facts, the evidence. Comments are bound to correct me. I truly dread to point it out but these steps do become subjective since it is up to what one wishes to observe and what one most expects to see. The subjectivity is meant to be rinsed away by the careful practice of double blind, mature, earnest record keeping. It is not easily done nowadays.

Since math and chemistry and physics have become mere starting points for modern science we like to think of science as being as accurate as these. In my opinion once you leave them behind science loses correspondingly more credibility. If you study what happened to string theory [see Not Even Wrong, or The Trouble With Physics] you will find that even physics has yielded to the most subjective and passion based waves of fashion. Once scientific results motivate political discussion the game is truly up.

I can well imagine that the well intentioned goal of this movement will be to lead, direct or embarrass non-members into accepting a set of facts that make the members feel right and justified. Good for them. They know what they want and should certainly have it. This is a long way of strongly recommending that you must use Neither Science or Logic your title.

Instead the proper term to use is Evidence Based. Here the quality of the observation is open to consideration by all, not only to the scientist observing. And the conclusions are also open to the more than just the members interpretation, not as the final product of the institution whose survival depends on the results to maintain its budget or else just hang it up.

Others tell me it is not up to Josephine six-pack to understand why a certain Chi-Square distribution is or is not appropriate for a given set of data. I agree, somewhat. My reasons of why and why not belong in the comments. Here I have answered the question and explained why it fits.