In LO Writer, Tab never increases indent. In the three cases you describe, Tab is just considered as a character and is inserted or replaces the selection.
There is a special context in which Tab behaves as a control command: llists.
Other document processors do not manage cleanly styles. At best, they know about paragraph styles. LO Writer has more categories: character, page, frame ans so-called “list” styles (in fact, they should have been named “sequence” style).
When you combine a paragraph style and a “list” style, the resulting paragraph style is fit for formatting list items. In such paragraphs, with the cursor set at the very beginning of a list item, when you hit Tab, you increase the item level within the list (going from level-1 a; to level-2 a.a. and so on when you’re already at level-n). Reciprocally, Shift+Tab will reduce the item level.
This could visually change the indentation of the list item but Tab is not the key factor for that. Level indentation is defined by the list style (the category I named “sequence”) where you can set how each level is indented and how it is numbered (labelled).
(For lists implicitly created with the toolbar button, settings can be changed with Format>Bullet & Numbering (an implicit internal style independent from any custom style).)
As you can see, the semantics is quite different from other document application.
Note, by default, Ctrl+M is Format>Clear Direct Formatting, not increase indent (unless you customised your keyboard shortcuts).
For elaborate documents, I highly recommend to work with styles and no keyboard formatting equivalent. It is much better to separate content and appearance with an adequate and consistent use of styles. Direct formatting is a source of insurmountable problems. Usually you have a preferred aspect for your document. Translate this aspect into styles and stores them in a template which you’ll define as your default. Then all your documents will have the same look and you’ll type faster because you no longer need to manually format.
Of course, this is only a partial answer to your question.
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EDIT 2020-04-16
Remember that Writer is not Word and does not pretend to be a drop-in replacement.
Writer is different from Word and explores other ways of formatting text. Of course, the bases are shared by all programs but how it is implemented varies. One of the highest-valued progress in Writer is the very clean definition of styles.
Nolens volens, any change of tool requires an effort to accept the differences. This is the hardest psychological part, all the more M$ threatens the common user with their “FUD” (fear, uncertainty, doubt) propaganda.
However, you can usually emulate Word behaviour with Tools>Customize, Keyboard` tab by assigning key combination to an action.
In your case, commands Increment/Decrement indent value are available. But, the Tab key (with or without other modifiers) can’t be used. You might use another key but your users will have to get used to it.
Note that you can customise key bindings and install it on all your PCs so that all machines behave identically.
