Meaning – Some vs. Certain: What’s the Difference?

In this context "certain" means a predefined subset. Usually it's used when the subset is fairly arbitrary, and to shut down discussions on how that subset is defined.

If the infinitive form was used after verbs that began with the letter "d", then the book would simply say:

The infinitive form is used after verbs that begin with the letter "d".

But because the set of verbs where it is used is fairly arbitrary, and generally only learnt through experience, the book probably doesn't want to delve into the rules, preferring to list a few common examples, and say:

The infinitive form is used after certain verbs.

I might say "my party officially starts at 9pm, but I've invited certain friends round for dinner at 7" if I didn't want to get into the details of which people I invited to the pre-party dinner party.

^∗ I'll watch some movie.
^okI'll watch some TV.

The difference in acceptability of these two statements turns on the fact that (at least in the US) different sorts of activity are involved.

When we watch a movie, we ordinarily watch the entire movie. If we're watching it at home we may interrupt the viewing from time to time to fetch snacks or go to the bathroom, but unless the movie turns out to be boring or disgusting we usually consume it as a complete story, from beginning to end. In effect, movie is a count noun, and watching a movie is an 'accomplishment', a telic action with a marked endpoint; so (at least in US idiom) it sounds odd to speak of watching some movie.

But when we (in the US) watch television, there is often no intention of watching a specific program. Watching TV tends to be a time-filler, a 'browse': particularly since the proliferation of cable/satellite channels into hundreds, we often turn the TV on to see what might be interesting, and we watch until we are satiated or bored or have to do something else. In such a context, television is a mass noun "with the syntactic property that any quantity of it is treated as an undifferentiated unit" (Wikipedia), and we may very well speak of watching some TV. It means we will watch TV for some time, for an indefinite while.

Note, by the way, that neither the awkward watch some movie or the acceptable watch some TV employs some in the sense which it has in many non-'Anglo' Englishes, as a indefinite determinative alternative to a/an—the sense which appears to be in play in your daughter's followup suggestion of I'll watch some program on TV.

This use is very uncommon in AmE, and I believe in BrE as well. Both colloquially and formally, some is employed in this sense only to mark its head as "definitely indefinite". For instance: If I tell you There's a guy here to see you, you may ask me Who?; but if I tell you There's some guy here to see you, you will not ask, because my use of some tells you that I don't know who he is—he's just some (unknown) guy. In the same way, if my son tells me that he watched some movie in his history class today I can be pretty sure that he did not pay very close attention and will not have anything interesting to say about it.