I don't doubt for a minute that Williamson would have referenced the strength of backbench feeling on the issue and the extent of Tory backbench support for more defence spending - why wouldn't he. Beyond that, well yes it is the Daily Mail and they have papers to sell and they've done a good job of embroidering it into a much more inflammatory headline.
The "Williamson vs Hammond" angle being pushed by the papers is unhelpful though, as a good outcome for defence would be portrayed as a Hammond defeat. And Williamson presumably did himself and the whole thing no favours by roundly pissing off the Chancellor early on, by embarrassing him over the payment for MoD flights issue, just when he needed his goodwill.
We can only hope that those taking the decisions can rise above it all, but I'm not holding my breath.