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For U.S. residential real estate (permits, starts and completions), how are single-family and multi-family buildings defined? How do condos and co-ops relate?

The definitions of the U.S. Census Bureau for multi-family and single-family are used, which (a) consider only the physical structure, not the ownership, of a residential building, and (b) count the individual housing units.

A housing unit is a house, an apartment, a group of rooms or a single room intended for occupancy as separate living quarters.   Separate living quarters are those in which the occupants live separately from any other individuals in the building and which have a direct access from the outside of the building or through a common hall.

The single family category consists of:

• Yes: All detached one-family houses.
• Yes: All attached one-family houses separated by a wall that extends from ground to roof with no common heating system or interstructural public utilities. 
• Yes: Prefabricated, sectionalized, panelized, and modular homes which are manufactured partially off-site, but which are transported and assembled at the construction site.
• No: Mobile homes.

To expand, the attached single-family category consists of housing units that are separated by a ground-to-roof wall, are not stacked vertically, have separate heating systems, and have separate utility meters.

Conversely, the multi-family category includes all buildings containing at least two housing units which are adjacent vertically or horizontally.  If built side-by-side, they (1) do not have a wall that extends from ground to roof, or (2) share a heating system, or (3) have interstructural public utilities such as water supply/sewage disposal.

The U.S. Census defines separate multi-family categories for buildings housing two families, three-or-four families, or five-or-more families. 

Each housing unit (separate living quarters) within a multi-family building is counted separately.   Therefore, a single structure classified by the Census as a multi-family building containing 50 units would add 50 to the total counts of multi-family permits, starts or completions.

The U.S. Census (and Moody's Analytics) estimates do include condominiums and cooperatives, insofar as they can be classified as single-family or multi-family structures -- they are not identified separately.  Condos and co-ops are kinds of home ownership, not structure.  You may as well ask "Do the stats include residences with flat roofs? With yards? With paid-off mortgages?"  They do, but those attributes are irrelevant to the dataset.