No. The wired stick up cam is always-on, recording to a buffer just like the flood light cam does.
Battery powered devices (and this includes battery devices like the Ring 1 & 2 whether wired to the doorbell wiring or not, and Spotlight Cam Solar, which even with the external power sources, are still battery devices and operate as such) are always asleep except for the passive infrared motion sensors, which will wake the camera and then begin recording. This leads to the delay.
Wired devices like the Stick Up Cam Wired will be just like the Floodlight Cam in how the motion detection and recording works, and will capture the initiating event, unlike battery models.