Let me add my voice to the chorus saying that "Stack Overflow" is the wrong brand to standardise on for other products.

As a case study, I would refer you to Mozilla, whose most recent rebrand announcement discusses why they are moving in completely the opposite direction:

Our past experiments with a combination of brands across a variety of products demonstrated that positioning these offerings under Firefox often created confusion, as Firefox was so closely associated with browsing. By aligning these initiatives directly with Mozilla, we’re simplifying the message. This approach also gives us more flexibility: Each product can better communicate its unique value without the baggage of misaligned expectations.

I think we can translate that directly to your brands:

Positioning Q&A sites for topics such as Photography, Cooking, and Interpersonal Skills under Stack Overflow will create confusion, as Stack Overflow is so closely associated with programming.

Simultaneously, you're proposing to move some services slightly away from the "Stack Overflow" branding - "Stack Overflow for Teams" is to become "Stack Internal". This seems like a good move, and a better model to follow across the board.

I frequently see people referring to each site as "a stack" (e.g. "the photography stack"), so just make that usage official. Adding the word "overflow" to that doesn't really help anyone - it means nothing to a photographer.

Let me add my voice to the chorus saying that "Stack Overflow" is the wrong brand to standardise on for other products.

As a case study, I would refer you to Mozilla, whose most recent rebrand announcement discusses why they are moving in completely the opposite direction:

Our past experiments with a combination of brands across a variety of products demonstrated that positioning these offerings under Firefox often created confusion, as Firefox was so closely associated with browsing. By aligning these initiatives directly with Mozilla, we’re simplifying the message. This approach also gives us more flexibility: Each product can better communicate its unique value without the baggage of misaligned expectations.

I think we can translate that directly to your brands:

Positioning Q&A sites for topics such as Photography, Cooking, and Interpersonal Skills under Stack Overflow will create confusion, as Stack Overflow is so closely associated with programming.

Simultaneously, you're proposing to move some services slightly away from the "Stack Overflow" branding - "Stack Overflow for Teams" is to become "Stack Internal". This seems like a good move, and a better model to follow across the board.

I frequently see people referring to each site as "a stack" (e.g. "the photography stack"), so maybe just make that usage official? Adding the word "overflow" to that doesn't really help anyone - it means nothing to a photographer.

PS: You might also learn from this retrospective of Mozilla's previous rebranding exercise.