Both of your options have significant CO2 emissions, so they are a no-go in just a few years.
Liquid methane is essentially the same as LNG, which is rapidly becoming the most popular fuel for newbuild ships today. But it's about as environmentally friendly as building natural gas powerplant to replace coal - a temporary solution at best.
Future solutions need a carbon-free fuel, period.
There's nothing wrong with CO2 emissions, as long as they remain carbon-neutral. So if you capture carbon dioxide from the atmosphere, and then use it to synthesize methanol or methane, then there are no problems with that.
Methanol is slightly preferred because methane can leak, and it's a more potent greenhouse gas than CO2. However, even most of the CH4 leaks happen near the drilling wells, and in pipelines. It's unlikely that synthetic CH4 will have to be transported over long distances.