'A character can spend Fate to alter the outcome of a check by either adding their Spirit die or re-rolling the original roll; make the decision to use Fate after the original die roll. The character may do both, once each, but no more.'

I had to re-read this 3 times before I got it. Suggest:

'After rolling dice, a player can spend 1 Fate to re-roll one die. Only 1 re-roll is allowed. He may also spend 1 Fate to roll his Spirit die and add it to the RN.'

Actually, if I have advantage and spend to re-roll, do I roll 1 die or both? You should specify in this section.

'Fate cannot alter a Critical Failure ...'

Can spending Fate cause a Critical Failure? If I add my Spirit die and roll a 1 on it, is that a crit fail? If I spend Fate to re-roll a 3 and roll a 1, did I shoot myself in the foot? I'm guessing no, but you should state it clearly.

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Thinking of RPG style, if I wanted to tie your mechanics into the story more, this is where I'd do it. The player gets to spend an important, finite resource tied to improvement to try to turn failure into success. With the Rule of Three, you could spend up to 6 Fate trying to influence one outcome! This is how a player declares, 'this thing in the fiction is really, really important to my character.' There's a real cost in the mechanics, you just need rules to tie the expenditure of Fate into the narrative. What does it look like/feel like when a player spends Fate? What kinds of things do we say? Who says it? How do NPCs (or even the environment) respond to spending Fate?

You may be doing some or all of this at your table, you just need to make your design and intent clear in the rules.