Normally, the problem with the precognition of major events is that you can’t change them. Even worse, by predicting them, you often make yourself a suspect. However, since the OP has mandated that I can change the things that I foresee, precognition would be the best choice to me.
Precog all the way–I already pick up more than I’m comfortable with just on my own intuition and reasoning and I sure as shit do NOT want more of that. Animal telepathy, sure, but I don’t even want to TALK to most humans, let alone know all the nasty wormage happening in their heads. Nightmare fuel, that.
Precog. Then that hot looking weathergirl is going to get hooked on me. 
Telepathy the way the OP describes it would make you a perfect assassin.
I’ll take precog, thank you.
My reading of that is that the future is not immutable, not that you have unlimited ability to change it. So if you precog that you are going to get hit by car, you don’t step off the curb. Future changed.
If you precog that Russia is going to nuke Omaha, you can try and get the ear of Putin and tell him not to, but no guarantees.
I think precog is less dangerous to my morals, as others have noted. What I think would be fun would be becoming the world’s best baseball player (assuming some talent). Always take that extra step to the right so that you catch the ball; never swing at a bad pitch, never get picked off. Lots of walks, assuming you can at least learn how to foul off pitches.
Since it doesn’t specify “other” minds, I suppose one could telepath-wire themself to be benevolent in their powers. Or, at least, to eat better and exercise more.
Ooh, had never thought of that aspect. I’ve often thought “So’n’So would function so much better if someone could just tweak that ______ wire in their brain.”
It often being the self-esteem wire, or the empathy wire.
I’d go with precog - win a modest lottery (I don’t want to attract too much attention), set myself up comfortably, play the market carefully so, again, not to attract too much attention.
Then I can spend my time engaged in other projects I find worthwhile, with some ability to foresee consequences better than anyone else.
It IS gonna suck when I get to the end of my life and realize I’m done in 30 days or less, but perhaps I will be wise enough to tidy up loose ends and say good bye to my loved ones.
Suppose I choose precognition, and evidently so does Broomstick. We both foresee that we’re going to each be driving and I’m going to hit Broomstick’s car. So Broomstick decides to change the future and stay home that day. I decide that I will be able to avoid the accident, so I leave home but then I never even see Broomstick’s car. If the future can be changed, how can there be precognition?
I am thinking that what is being proposed was a “You, and you only” deal.
I could probably rewrite it so that it was just me. It was more of a thought experiment exploring the idea that the future can be changed. If I have a precognition about the future but then I can change the future, my precognition did not show the future. I am having a hard time accepting that the premise of the OP is internally consistent.
The OPs definition of telepathy is basically mind rape and I’d be horrified for anyone to have that ability, including me.
I read a SF short story where the main character could do that and it totally creeped me out. I thought I didn’t want to know the future, but between the two choices, that’s the one I will go with because reading and controlling minds is not cool to me.
I believe the way the OP meant to reconcile this is a choice web. You could choose to turn right or turn left but either way the program knows what will happen next. In this case you can see the web for the world for the next 30 days.
You would know that it’s 80% probable that Russia will nuke the US in the next 30 days and then like Dr. Strange you could pick out the path of events that would be required to guide that probably in the direction you want. The closer the event the more it would be locked it. Tomorrow’s stock market is 90% locked in but if you placed a trillion dollar short on the market as a whole you could drag it in a different direction on the other hand your $100 short would move it very little.
Ohhhhhhh, if the telepathy works on dogs and I can make them forget how to bark, then I pick that.
You (and I) can see the possible/most likely futures and take action to choose the one we find most favorable.
Precognition, hands down. But having precognition implies I have cognition, does it not? How could I know more about the future than I know about the present? To stick with the example of the lottery, repeatedly cited, I ignore last week’s winning numbers. Of course I know where to look them up, but I don’t know them, I guess most people don’t either. So how do I know next week’s winning numbers? Those I sure cannot look up. What kind of knowledge is that? Same applies to the stock market, the other recurrent example here.
And telepathy (which I do not want or need, if I understand it correctly) seems to imply that people have their thoughts and ideas well sorted in their minds. Well I for one do not. Most of what is in my head is either subconscious or deeply stored. I often don’t know what I mean before I said it, if at all. I seriously doubt anyone could make heads or tails of the chaos in my head and I would like to think this applies to everybody else too.
Edit: So I am choosing not precognition, but cognition. I would be happy if I knew something at all with some degree of certainty.
If you can see the future, you can see a future where you look up the lottery numbers. You can see a future 10 years from now where you look up the ‘best performing stocks of the last decade’.
I don’t want to turn into a threadshitter but I think this is an interesting sidebar. This notion of seeing multiple possible futures suggests that what we have here is not precognition. I make decisions every day based on possible futures and take action to choose the one I find most favorable.
You are making decisions based on likely outcomes. If you are a precog (and especially if you are the only precog (or at least a limited number), then you are making decisions based on known outcomes (two-headed coin, so to speak).