If I do not pay the monstrosity price, but instead have a different spell or permanent give the same amount of counters to the creature, is the monstrosity ability activated?
I feel really sorry for you: waiting for an answer for 15 minutes was inconceivable here, two weeks ago.
But the website has been turned upside down lately; WotC called this «improvement». Go figure.
Wizards of the Coast: outsourced to Elbonia
If you're old enough, you'll remember that this is how the Internet looked like, (and functioned like) 30 years ago.
To answer the question, no.
Adding +1/+1 counters will not trigger the monstrosity trigger.
The triggered abilities that trigger on a creature becoming monstrous only trigger on them actually becoming monstrous. This doesn't have anything to do with the counters that are being placed on the creature with Monstrosity. The Monstrosity ability has two independant effects : putting a number of counters, and making the creature monstrous. One isn't dependant on the other, you can activate Polukranos, World Eater's ability with X = 0, and it still becomes monstrous (for what it's worth, i.e. not much :P). Putting counters on the creature by other means doesn't make it monstrous.
you can activate Polukranos, World Eater's ability with X = 0, and it still becomes monstrous (for what it's worth, i.e. not much :P).