Normally that's fine, because the tab stop points to the next field. That's the whole point of tab stops― you place them at the beginning of fields on a forms. (Yeah, you also need to be on the right line, so a single key press rarely gets you there, but we were working with purely mechanical controls.)

Where it goes off the rails is when we embed a user interface inside another user interface. That's not a problem specific to the tabulator, but also with all novigation functions, like home, forward, and back.

I rarely use web-based email interfaces, but when I do, I accidentally send half-written email messages, because I'm trying to use a block quote or something similarly indented. It's especially bad when the next field is the 'Send' button, and pressing the spacebar sends the message.

"Next field" is sort of what tabulation means. Moving to the next column of a tabular form.

That was the origin, but for everyday typing of non-tabular work, like letters, it was used for indenting paragraphs and address and signature blocks. And it always moved to the next tab stop to the right on the line. ASCII calls it HT, ‘Horizontal Tabulation’.

Later, some systems, notably IBM's, overloaded this to move to the ‘next’ field on a form, which might be to the left and down the page. So from there MS-DOS/Windows stuck us with a Tab that sometimes moves horizontally within a text field, and sometimes moves to a different field. Just like Return sometimes moves to a new line/paragraph and sometimes submits a form.

That is not an overloading, that is just preserving the behavior of Tab on IBM terminals.

Need an "indent" key distinct from the "tab" key.

Yeah, maybe something that produces a fixed size space and you press it multiple times until your preferred indent.