What is the difference between these 4 words for the definition of "expectation":
Specifically, which would you use for "managing customer expectations (for project outcome)"?
"お客さんの期待" Customer Expectations.
期待 will be the one you're looking for for "customer expectations". It's what you use if you're looking forward to something, what you're "hoping to see". Used for "fulfilling expectations".
予想 is more "neutral" in that it has more to do with a way you predicted something to turn out. "I expected it to be this way". Close in meaning to 推測.
予期 means according to my dictionary to a correct expectation, or a correct prediction of something ahead of time. Both examples given were negative: 「予期せぬ出来事」 (which seems to be a set phrase) "Something we did not predict/expect", and 「予期に反した結果」 "A result contrary to expectations". It appears to be more formal and overlaps in meaning with 推測 and 期待.
思惑 refers more to thoughts or perceptions on something, but also specifically economic predictions. "Speculation" might be the best word to describe it. Where as the three above can be used as する verbs, it is very uncommon to use 思惑する. In fact it is an ateji of an old nominalization grammar structure: 「ク」, so to turn it into a verb you would just use 思う.
Hope this helps.