While I agree with you about the permafrost cows really do burp more methane than they fart. The reason for that is when cows are out grazing they are doing minimal chewing and just passing it to the first of their four stomachs, the rumen. In there the grass is microbialy fermented which is what produces the methane. When the cows upchuck the fermented grass to chew their cud they burp the methane.
If you're wondering why the cows need the fermentation it's a process that breaks down the cellulose in the grass they eat into more nutritional things like sugars that the cows can digest.