Question: What should be my response, when I have offended another on account of their personal convictions, and not my own?
You should ask their forgiveness. And if you have looked down on them for their preferences and judged them in your heart, you should also repent before God.
In Romans 14-15, Paul is talking about preferences and personal convictions over non-biblical issues. To “the strong” – those whose conscience allows them freedom – Paul says, “now we who are strong ought to bear the weaknesses of those without strength and not just please ourselves. Let each of us please his neighbor for his good, to his edification” (Rom 15:1-2). If you have offended someone in an area of personal conviction (and not biblical morality), then you should ask their forgiveness and resolve to live around them in a way that shows grace towards their weaknesses.
There’s much more that could be said on this issue. But rather than give you my thoughts, let me simply encourage to read 1 Corinthians 8-10 and examine what Paul says about WHY our liberties exist and then read Romans 14-15 which describes HOW to get along in a church where people with different convictions coexist. This is not a new problem. Scripture speaks amply & ably to it.