So you can run both "Clicks to Website" and "Page Post Engagement" ad types in the newsfeed under your page.
Only difference it seems is that with the latter you have to enter an URL (eg. bitly redirect) with a call-to-action in the ad copy while as with the "Clicks to Website" option the user get's redirected after she clicks on either the ad picture or title. Optionally Facebook provides some standard CTA buttons like "Learn more" that you can include.
CTR%-wise, which you do you prefer?
From my experience I have noticed that the overall click to website cost is lower on the page post engagements assuming you can get your 2nd CTR up enough
You're conflating the campaign objective with the post type.
With a link post, the image displays as a wide rectangle and a click on the image goes to the website. With a photo post, the image is more "square-ish" (depending on whether it's displaying on desktop or mobile) and a click on the image opens the enlarged image in a light box.
Both post types can be used with either campaign objective, although photo posts are most commonly used with page post engagement.
John is correct.
Campaign Objective can basically be ignored by affiliate marketers.
It locks the ad groups in the campaign to specific ad types, but this is only so that people with very little knowledge make ads most relevant to their objective.
If someone wanted to get mobile app installs and you let them make photo posts... they probably would and would get confused > complain and wonder why their apps aren't being installed.
So, whether you have website clicks, page post engagement, or website conversions as your objective, a page link post within that campaign will be exactly the same.