Is acceleration noise modelled differently in EKF and UKF Kalman Filters?
Absolutely! 

Every (extended / unscented) Kalman filter starts with a signal model.  The state update equation in the second image is a very different beast from the little you show of the first model in the first image.

Without seeing the complete model, it's hard to say whether it's all correct, but if the signal models are different, then certainly the state update and output equations for the filter will be different.

For example, this paper shows two EKFs for frequency tracking that use two different signal models.

Regarding incorporation of the $\Delta t$ terms in the acceleration, that does seem a bit odd.  Usually those terms form part of either the state transition matrix or the input matrix. See this answer for one way to do that for a simpler model.