Your mother hadn't made it clear that she would like your DS to mow her lawn. If it is to be a regular thing, it should be set up as such and explained to DS that he needs to go round and wield the Flymo weekly.
Pegging the lawn-mowing to birthday/Christmas gifts is nasty of her, especially after the event. If it's to be a regular thing, it's either just one of his expected chores or it's done for a fiver each time or something. Saying nothing, building up resentment, and then punishing him by withholding gifts is... well, it's unpleasant and manipulative, isn't it?
When you sided with your mother telling her that it's a struggle to get him to do anything effectively you've told her that you think he's lazy -- and also that it was OK by you to withhold gifts. That feels a bit mean, although I can appreciate you felt caught on the hop and stuck in the middle there. You effectively agreed with her stance on the whole thing and cut your son off at the knees (metaphorically speaking) to appease your mother.
And sure, your DS could be a bit more helpful. But he's also 14 so might need some help at least setting up the frameworks for the helpfulness.
Is there a wider context here, about how your mother is? If so the best way forward might be to sever the link between DS doing things and her 'favour' now rather than later. If they needs the lawn mowing, they either do it themselves or they get someone in to do it. Making their expressions of love for your DS conditional on the performance of a task that they haven't made clear is manipulative.