grep date from 2nd line of a text file

As always, there's more than one way to do it, depending on the exact requirements.

As one of the simplest ways, you could do:

AVARIABLE=$(sed -ne 's/.*DATE://p' document_determinationv1.txt)

This doesn't actually care which line the date is on. It will work as long as your file contains exactly one line containing the keyword DATE followed by a colon and the date you want to extract. It will also not check whether DATE appears as a word of its own, whether the part after DATE: is actually a date, or whether it is followed by any additional junk. If your file contains more than one line containing DATE followed by a colon then it will put all of the parts after each DATE: into the variable, separated by newlines, which may or may not wreak havoc with your further processing. So if your file contains, for example:

[DOCUMENT_DETERMINATION]
#REVISION:v1;DATE:20210805;WEIGHT:123kg
[REAL_DETERMINATION]
#PRIME:13;CANDIDATE:BART SIMPSON

then the command will happily put

20210805;WEIGHT:123kg
BART SIMPSON

into the variable, including the newline between the letters g and B.

But as long as you can guarantee the file has exactly the format you quoted in your question it will work fine.