Politics: the activities involved in acquiring,
using, power in an institution that influence or are able to influence decisions that affect a country or a society.
Now, I could be mistaken, I’m not omniscient, my memory can and does fade, but I cannot recall anything Jesus is reported in the Gospels to have said or did that qualifies Jesus as being political or engaged in politics.
Furthermore, Jesus’ teachings, which arguably, if followed by everyone, has political implications, isn’t the same as Jesus being political or engaged in politics. Just as Aristotle’s thoughts of a First Cause, an Unmoved Mover as the cause for everything may have implications of a religious kind, doesn’t render Aristotle engaged in theology.
OK, even using the definition you have selected:
Israel's politics and religion were inseparable. All that Jesus dealt with among the Jews thus involved both. Jesus was sent to deal with Israel, not Rome. Now He is King of Kings and all the earth is His inheritance. What is He not Lord over?
Our Faith, even though most would like to separate it from politics for obvious reasons, cannot really be separated from the activities or effects of human government. Our Scriptures instruct us on our morality, our ethics, and much more. Human Government has been identified as a Divine Institution from Genesis. Paul's instruction in Romans 13 re: submitting to authorities is stated in the context of the authorities being God's servants to reward good and punish bad. How do we separate ourselves from living under governing authorities, which "influence or are able to influence decisions that affect a country or a society" especially in a nation where we have the rights we have to voice and vote godly morality and ethics and legislation? Buying a party-line and quite possibly shirking our responsibilities in Christ does not mean that party-line is correct.
John dealt with the issue of the Logos. Apparently he viewed it as theologically relevant and thus the philosophers engaging themselves in theology, whether intentional or not. Since this is God's creation, what is not theological?
I just answered in another thread a misrepresentation of Jesus re: the "Golden Rule." I don't think the poster wanted to argue theology, nor, given how he identifies himself, do I think he even thinks theology is a real topic. He's just arguing to keep Christian thinking out of "secular" government.
Since every major human institution, including human government, and thus politics, is based in Scripture, at this point I don't see these clear and customary demarcations anymore. I'm not alone in this.
Since the OP is asking if Jesus is a politician, maybe we should be defining "politician." One definition I looked at quickly said something about politicians being involved in legislating. I wonder if God is a Legislator and whether or not Jesus was involved in debating Law when here, or whether He still is involved in Law and will be when personally here again.