I am not sure if this bicycle fits in this 'cross frame safety' department. But it isn't a diamond frame either.
There seems to be nothing like this Quadrant no. 13 ...
It is one of the first models of safety bicycles, introduced in 1887 and built at least until 1889, maybe 1890. The very strange looking steering mechnism really was the usp of this model. Quadrant had introduced it on their tricycles a few years earlier - the Encyclopaedia of Bicycle Manufactureres (V-CC, Ray Miller) calls it 'a revelation in smoothness and efficiency'.
Clearly, Quadrant was proud of this system and used it on their earliest safeties, like this no. 13, built in 1889-1890. Like no other bike it shows the struggle for a perfect bicycle. Isn't it incredible that in 1892 there was no discussion any more among producers and bicycle-journalists about frames? They all agreed on the perfect frame: diamond shaped with straight seat tube. We still agree on this.
You can admire this fine experimental bicycle in the Velorama museum in Nijmegen (Netherlands), alongside with other very early safeties, like the BSA, the Rover and the Humber.