Pressing tab to indent a list moves to the next table cell

Solution 1:

Shift+Alt+Right Arrow Key will demote your bullet inside a table, whilst Shift+Alt+Left Arrow Key will promote your bullet.

Solution 2:

This is not, strictly speaking, an answer to your question, but rather a solution to your problem.  You can increase the indent by clicking on the “Increase Indent” button in the “Paragraph” group on the “Home” tab of the ribbon:

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If you have musophobia (fear of mice (and rats)), you can use Alt+H (if necessary, to select the “Home” tab of the ribbon) and Alt+AI.
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Solution 3:

As far as I know there's no way to turn this off entirely, but the best method I've seen is the one given by Microsoft directly.

To indent inside a table cell, click in front of the text or numbers you want to indent, and then press CTRL+TAB.

Solution 4:

When I have had to enter indents into tables I have always done this:

Enter some obscure text where you want your indents (Example: "TABTEXT")

Find and replace that text with a TAB by entering "^t" in the replace with box.