I'll add something else: "I like to think ___" can mean you prefer to approach a situation optimistically. "I like to think people are good, until they show me they are bad." "I like to think that everyone is acting with the best motives. It helps me be more understanding." If you say "I would like to think ___," then you're showing uncertainty. "I would like to think the best of everyone, but I've been cheated before." "I would like to think you know what you're doing, but I need to see some more explanation here."

Neither. It means "I hope things are this way" or "I wish things to be this way, although I have a doubt they truly are". I'm pretty sure this is the meaning you're looking for!

- I like to think first before signing a contract. - In this sentence "like" is used to show a person's preferred way of doing things. - I would like to think about it first. - In this one "like" is used to make a polite request. I hope this helps.