Initialize the eight working variables SHA256
The variables a, b, c, d, e, f, g, h are assigned on each round of the compression function main loop, but the interim hash values are considered only per message chunk (i.e. after all rounds have completed)

I found the Wikipedia pseudo-code easier to understand than the description in your question, and it is clear there how the variables relate to interim hash values. Note where your reference relates previous hash components $H_0^{(i)}=a+H_0^{(i-1)}$ the pseudocode just assigns h0 = a + h0, because there is no need to keep previous hash values, just a current one which is updated after processing each chunk.

Just before calculating the rounds for the each chunk, both the current H array and a, b, c, d, e, f, g, h hold the same data. This changes during calculation of the rounds, and you need to keep the value of H so it can be added back in at the end of the chunk.

When starting the first message chunk, before any processing rounds, H and a, b, c, d, e, f, g, h hold the "initial values of H" (based on square roots of first 8 primes). You do not use those initial values again.