The definition is "read (something), typically in a thorough or careful way."

which meant that 1 is correct.

Strange, because I always thought that 2 was correct.

I would be interested to know why you raised this point as I appear to have been wrong for many years.

I thought it meant "to read thoroughly and carefully" and am surprised to learn of the drift in meaning. 

ahdictionary says:

https://ahdictionary.com/word/search.html?q=peruse

1. To read or examine, typically with great care.

2. Usage Problem To glance over; skim.

They have one of their usage notes:

Peruse has long meant "to read thoroughly," as in He perused the contract until he was satisfied that it met all of his requirements, which was acceptable to 75 percent of the Usage Panel in our 2011 survey. But the word is often used more loosely, to mean simply "to read," as in The librarians checked to see which titles had been perused in the last month and which ones had been left untouched. Seventy percent of the Panel rejected this example in 1999, but only 39 percent rejected it in 2011. Further extension of the word to mean "to glance over, skim" has traditionally been considered an error, but our ballot results suggest that it is becoming somewhat more acceptable. When asked about the sentence I only had a moment to peruse the manual quickly, 66 percent of the Panel found it unacceptable in 1988, 58 percent in 1999, and 48 percent in 2011. Use of the word outside of reading contexts, as in We perused the shops in the downtown area, is often considered a mistake.

Shrug. It is what it is. Now I know. Language evolves. 

if you perused a magazine you skimmed over it briefly = you per + used it.  = you pre used, pre read it quickly before either deciding to buy or not. (past tense)

If you perused any document you per+used it to quickly find something. You skim scanned its contents to find something.

If you say you will peruse something you are saying you will read something throughly. 

If you peruse something in the present tense. You read it throughly to find detailed information. 

"The lawyer perused the legal document before writing notes for his clients defence" 

This now becomes an ambiguous sentence; because we do not know if the lawyer quickly scanned the legal document, or studiously studied it before or whilst writing down notes, for his clients defence.

It is not a common word today. Like you say its meaning is now vague in modern English. Much would depend upon the context of the sentence, and any sentence would now need more information to make the meaning clear.

@Norman: because even dictionaries gives double definitions, like the Cambridge dictionary, that says:

peruse

verb [ T ] US ​  /pəˈruz/

​

to read or look at something in a relaxed way:

He opened the newspaper and perused the sports pages.

​

Peruse can also mean to read carefully in a detailed way.

@ Ishtar your search has obviously been more successful than my internet search. It does not seem to mention the latin, but it does say middle English per "completely" but it could be a different formation meaning casually from 19c. 

It was always used to mean "read throughly in my youth, and only in the past thirty years have I heard it used as reading casually. With misunderstandings as to which meaning is being used. 

My internet etymology search that I used to try and refresh my memory only indicated at the middle English pre (before) throughly with a reference to it possibly meaning casually. 

The lesson should be to always use two or more sources to cross reference. Which I was taught to do but failed to do so this time.