The IPv4 checksum does not add bytes one at a time. It pairs them into big-endian words and treats the checksum field specially during computation.
highlighted = computed this step
Why words are paired bytes
The IPv4 checksum reads the header as big-endian words. Each row in the table is a displayed word recomputed from the header bytes.
word bits=16
Ten big-endian words
The 10 words begin 0x4500, 0x003c, and 0x1c46. The table lists all of them in wire order.
0x4500,0x003c,0x1c46
The checksum word is zeroed for compute
For computation, word index 5 is treated as zero. That is the checksum field itself, shown in Book 1 as 0xb1e6.
zero word=5
Summary
Checksum arithmetic starts by pairing bytes into big-endian words and zeroing the checksum field for computation. Error detection over the exact bytes; timing/throughput is not modeled here.