Transformer model in Power Systems VS Coupled Inductors model
In the power systems model "m" does not standing for "mutual", it stands for magnetization. The magnetization current of a primary winding has absolutely nothing to do with mutual coupling. It's always best to consider the fuller picture of a transformer in my opinion: -



The coupled inductor thing in the middle is in fact the perfect power transformer and this has a coupling factor of unity. It also has infinite inductance and zero losses. All the other components around it represent the real-world failings of any transformer.

You can modify the model like so: -



Now the secondary side components are moved to the left of the perfect-coupler via turns-ratio transformation and, if you want to take it one step further, you get what you have in your 2nd picture but of course the "thing" that does the coupling (k=1) has been disregarded.