Steering Vector for Generic STAP
Yes, $a'(\phi)$ is a steering vector. In fact, the third equation (almost) describes a phased array that is capable of steering lobe patterns at different frequencies.

Each element of the array has a filter behind it which forms a weighted sum of (delayed) samples in Time. The same idea is applied one level above by now summing (delayed) samples of the received signal from the different elements that are in different points in Space.

In STAP, the adaptation in Time maintains the coherence of the probe signal (or carrier) against noise / doppler and other "structural" distortions and the adaptation in Space maintains the coherence of the signal along a specific direction by trying to minimise the covariance between different elements. Think of it as one convolution per element of the array(typical filtering in the Time domain) and one more ACROSS the same time instant (the output) of the element filters (Space).

BUT! With this configuration you can tune the Time filters at some sinusoid and then with the frequency of that sinusoid tune the Space filter to receive from a given bearing. To do the same thing simultaneously, you would need a Fourier Transform behind each element to decompose the received signal into a bank of sinusoids and then apply different space filters ($a'(\phi)$) to those outputs for specific bearings. In this way, you could have the array listening to some low frequency from the left and some high frequency from the right simultaneously. I feel that this is what is meant with "...the more generic case is that the array response on all channels is not equal (more like an array manifold) and is thus..."

Hope this helps.