Conceptually, what is the main difference between a sandwich (direct) ELISA and an indirect ELISA?

There is no conceptual difference; both names are interchangeable and both always detect patient antibodies directly using two enzyme-linked antibodies in sequence, one after the other, to amplify the color signal.

Sandwich ELISA detects antigen in the patient sample captured between two antibodies, whereas indirect ELISA detects the patient’s primary antibody binding to a fixed antigen and uses an enzyme-linked secondary antibody for detection.

Sandwich ELISA uses only a secondary antibody to capture patient antibodies on wells coated with enzyme, whereas indirect ELISA sandwiches antigens between two enzyme-linked antibodies to detect proteins by color change.

Sandwich ELISA detects nucleic acids using antibody-enzyme complexes as labels, while indirect ELISA directly measures enzymatic degradation of substrate by patient-derived enzymes as a proxy for infection status.