She almost laughed out loud at herself, thinking she could trust her instincts about people.
"laugh out loud" VS "laugh out loudly", which is correct? I looked up they are all adv.
You're right. They're both adverbs. However, you can only say either to laugh out loud or to laugh loudly. There is no such thing as the phrasal verb to laugh out in English. It just does not exist. And because of that alone, to laugh out loudly would be an incorrect phrasing. However, there does exist the idiom to laugh someone out of something, but it means something completely different. Look it up when you have a spare minute.
Out loud is an expression unto itself, by the way. It means talk, laugh, think, read or do whatever you're doing in a way that other people can hear. You can use it with all kinds of verbs that describe actions involving some form of oral interaction. Here are just a couple of examples:
He was reading the Bible out loud for the blind boy.
Say it out loud so that I can hear you.
When practicing your English, it's important to speak out loud.
I would like to add to Michael's fabulous answer, which is correct and I upvoted it.
The idiom "laugh out loud" has a cultural perspective that I can't prove with references right now because the Internet is filled with LOL references and I don't have the time to weed through them. (Memes...) But, here's the gist...
In many cultures it's considered rude to laugh at someone. And in many cultures (perhaps all but we of the U.S.), it is considered rude to be loud and boisterous. Thus, to "laugh out loud" suggests a either an impolite action meant to be disparaging of another person or a loss of control or decorum over an (apparently) outrageous situation.
Thus, in written English, if you were only trying to convey that someone's laughter had volume, you would say they "laughed loudly."
But if you wanted to convey the idea that they were responding to an outrageous situation or were rude, you would say they "laughed out loud."
https://books.google.com/ngrams/graph?content=laugh+out+loud&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Claugh%20out%20loud%3B%2Cc0#t1%3B%2Claugh%20out%20loud%3B%2Cc1
– Rob Apr 19 '18 at 19:51