Rather than carbon footprint and carbon emissions--which would not account for human emissions of the highly potent greenhouse gases without carbon--it would perhaps be better to cover all pollution prevention, such as through ecological footprint:
Two of the most important greenhouse gas emissions from agriculture include enteric methane from cattle and sheep, and nitrous oxide emissions from artificial fertiliser and dung additions to soil. Obviously N2O is not carbon-based, so you would not literally call it a 'carbon emission' despite being a potent greenhouse gas which would be included in a 'carbon footprint'.
Ecological footprinting in the form promoted by the Global Footprint Network uses the sum of the hectares of land required to provide raw materials for people and for absorbing our greenhouse emissions as a proxy indicator for our total environmental impacts. It has the advantage of breadth but addresses excessive resource use more directly than it reflects other impacts. The official GFN footprint component for GHG is based on the land it would notionally take to absorb the GHG by growing forests, if we did that, rather than the movement of ecosystem boundaries caused by climate change.