I would think the author of this used 'take' to imply a use of force or effort. Maybe not literal physical force but beat out a competitor.
Simply 'obtaining' doesn't imply this. No implication of motive or tactics used without directly expressing it. In general, it has an overall mellow tone.
I obtained the documents. (could have been given/handed directly to me.)
I obtained the documents by stealing them from the desk. (expressing the effort)
I took the documents from the desk. (not necessarily stole them but implies more force or effort.)
To "take off" in this context means to not go to your job.
To "rest" means to stop working for a period of time.
The ideas are related. Often a person will take off from work so that he can rest.
I can see how reading a definition might be confusing. I carefully used the word "job" above, but we often refer to a person's job as his "work". So you could say that "take off" means to not work, and "rest" means to not work. But we're using "work" here with two different definitions. In the first case we mean "show up at the office or factory where you have a job". In the second case we mean "exert yourself, perform a task which requires effort".
You might take off from work (i.e. from your job) to do a home improvement project. The home improvement might be much more difficult and strenuous than your job, so you wouldn't say you are resting. You can take off but not rest.
Or you could take a break while at your job and rest for a while. Maybe you get a cup of coffee, or chat with co-workers about football. You are not "working" in the sense of exerting yourself, but you are "at work". Or maybe you're just lazy and although you show up at the office, you just sit around and chat on forums. (Oh, wait ...) You can rest but not take off.
Note that "take off" has (at least) two other totally unrelated meanings: It can mean to remove something, often an article of clothing. "Please take off your hat when you enter the building." It is also used to refer to the launching of an airplane or spacecraft. "The airplane will take off in 10 minutes."
Best Answer
Both of these are informal expressions and have multiple meanings:
To take on means:
to attempt to overcome someone or something (e.g. an obstacle or challenge as @DanTheMan says) - I'm going to take on the trail today.
to assume the duties or responsibilities of something - I'm taking on the manager's position.
to assume a specific demeanor or look - His face took on a look of disgust.
To take up means:
to pull or cause to rise upward, particulary with clothing - Can you take up the dress in the back a little?
to clear off or put away - Take up the dishes from the table
to start learning about a skill, hobby, or profession - Jon took up boxing and has been enjoying it.
So really take on is more appropriate, although take up could be used colorfully to express that John is somehow "rising upward" towards the menacing stranger, in a sense.
Also consider this: