Learn English – What’s the difference between “diary” and “journal”

Cooperating means working with someone in the sense of enabling: making them more able to do something (typically by providing information or resources they wouldn't otherwise have).

Collaborating means actually working alongside someone (from Latin laborare: to work) to achieve something.

The confusion comes from the overloaded meaning of "work with": In the "Work with me, people" sense, it means to go along with my idea - it's a passive condoning or suspension of disbelief rather than an active involvement. In the "I'm stuck, can you work with me on this problem?" sense it is a request for active commitment.

So in terms of helping achieve something, the ordering is something like collaboration, then cooperation, then passive indifference, then active obstruction.

'Lover' implies a sexual relationship. 'Love' does not, at least not necessarily.

You could call your lover your love with no problems arising. But if your love is, say, your child, you wouldn't want to call your love your lover!