Dictionaries characterise this idiom as less literal, as can be seen here:
If you say that you wouldn't mind something, you mean that you would quite like it.
- I wouldn't mind a coffee.
I wouldn’t mind (doing) something: used to say that you would like something
- She’s gorgeous! I wouldn’t mind looking like that!
Presumably this expression is listed in dictionaries because the meaning is not literal – otherwise it wouldn't be a separate entry, and one would simply have to look up the verb "mind" to understand all of its subtle nuances.
The Longman Dictionary sentence comes across as an example of litotes, and it is in fact saying "I would be [perhaps more than] happy looking like that".
The example I'm curious about is this one:
A: Would you like to go on a trip with me?
B: I wouldn't mind.
In this example, I think the meaning leans towards more literal, but I can't say for sure. I don't believe person B is actually saying "I would love/like to, definitely", but rather "I'm fine with a trip, sure". Would the meaning depend on the tone if the sentence were spoken? The sentence is as I put it above in bold.
To sum up, I'm essentially asking whether "I wouldn't mind." alone indicates interest like "I'd like to.", or less so, like "I'm fine with it. I have nothing better to do anyway."