We realized we'd seen both variations before, but didn't know why sometimes we saw one and not the other. We set out to find the answer ourselves.
First, I had my colleague Jillian, who is has an iPhone, call me.
We wanted to see if it would make a difference if the person calling you was in your contacts, so I had another colleague, Steven, who also has an iPhone but isn't in my contacts, give me a ring.
Negative. Did it make a difference if someone with a non-iPhone was the caller? Our front page editor Matt checked.
Nope! We got the slider again.
At this point, we gave up and did what we should have just done from the beginning and Googled it.
Turns out this is a pretty common question and the answer is pretty simple: If your iPhone is locked when you receive a call, as mine was, you'll be shown a slider to answer the call, with no option to ignore it. If your phone's unlocked, you'll get the option to either dismiss or accept the call.