Problem solved. Here is the solution in case anyone else is wondering:
Since LF and CRLF both share the LF character, I set the line delimiter as LF (0x0A). This works correctly for extracting the full records (with the side effect of having one extra CR character at the end when CRLF is the delimiter).
One can get rid of the extra CR character using a dummy field to absorb the CR character or by using a map.
Notice that since the LF and CRLF delimiters have different lengths (1 and 2 characters respectively), I had to do a couple more changes to the schema to ensure that both are handled correctly.
In my scenario, each line record that was parsed contains 8 positional fields, so having an extra CR character at the end resulted in an error due to Biztalk expecting a certain length for the last field that does not account for the additional CR character. The solution is to increase the length of the 8th field (which is a Filler field in my case) by 1. However, in order to still be able to handle the LF line delimiters, make sure you set the 'Allow Early Termination' flag to TRUE. This way no errors are thrown if the last field is 1 character short of its assigned length (if the CR character was not included).