Is connotation important when talking with unbelievers about Jesus's message of salvation?
Absolutely. Heck, even when talking with Christians from other denominations, connotation is important. It's not uncommon for, say, Orthodox and Protestants to completely talk past each other, just because the terminologies and mindsets they each possess have a lot of differences, some subtle, some obvious. Let alone when attempting to talk to non-believers...
I feel many Christians who grew up in church use code words and expect people to know the meaning, or people who have been hurt by religion to just accept the message and I'm just like
I agree. The best way to get non-believers to accept the message of Christ is to LIVE that message, and show them what it actually
means to have Jesus Christ as our Lord, God and Savior, and demonstrate for them the spiritual and heart-centered side of the message. And if people have questions as to the intellectual side of the message, then you can use and define the code words (emphasis on define).
BTW, welcome to the forums, trumpet_guy!