Indeed.

It's as if people are being deliberately mischievous by writing both "the SIAI" (which should be "SIAI"), and on the other hand, "Singularity Institute" (which should be "the Singularity Institute").

Luke is probably confused by the fact that the organization is often called "Singinst" by its members. But that expression grammatically functions as a name, like "SIAI" (or, now, "SI"), and thus does not take the definite article.

The full name, however, ("the Singularity Institute") functions grammatically as a description, and thus does take the definite article. Compare: the United Nations, the Brookings Institution, the Institute for Advanced Study, the London School of Economics, the Center for Inquiry, the National Football League.

Abbrevations differ as to whether they function as names or descriptions: IAS, but the UN. SI(AI) is like the former, not the latter.

If the abbreviation is an acronym (i.e. pronounced as a word rather than a string of letter names), then it will function as a name: ACORN, not "the ACORN" (even though, in full, it's "the Association...").