There is no clear answer to this question due to the 'dual nature' of methane (biogenic and magmatic) - both of your arguments are right. Some things in nature just don't fit into the classification schemes that humans invent for them...

Both are right.

Good question, but it depends on the pathway of carbon assimilation the organism is using. To be considered strictly autotroph, an organism must fix all its carbon from inorganic carbon (mainly CO2) through the Calvin cycle or some other autotrophic pahway like reverse citric acid cycle, hydroxyopropionate or acetil-CoA , for instance. In the aerobic process, these bacteria oxidate methane to formaldehyde and then formate and finally CO2. All or 50% of the carbon can be derived from formaldehyde. In the serine pathway, there is indeed CO2 fixation with the formation of serine from formaldehyde and CO2 and then the organism behaves partly as autotroph. In the ribulose monophosphate pathway all carbon comes from formaldehyde, then the organism is behaving as heterotroph. Methane is an organic molecule, regardless of how it has been originated.

This has been discussed in the following short note

Article Methylotrophy versus heterotrophy: A misconception

This concluded methanotrophs are heterotrophs or at a push mixotrophs and its a bit complex. And suggested to differentiate using "polytrophs" to refer to non methylotrophic heterotrophs when making a separation between the two.