I don't think so, because the sweeping motion of minute hand is effectively continuous rather than discrete, so there's no truncation. At 4:53:30 the minute hand will be correctly in between 4:53 and 4:54, if one (like the author) cares about such precision.

Technically Big Ben is the bell, the clock has typically been referred to as the Great Clock because that's sufficient explanation of what you meant. The tower, which like the clock is often referred to just as "Big Ben" colloquially, was formally named "Elizabeth Tower" in honour of the late Queen some years ago.

Depends if the analog clock “ticks” on the minute or if it does a constant movement during the minute.

Not really. A mechanical analog clock will typically have smooth motion, so 30 seconds into a minute, the minute hand will be halfway between two values.

Most quartz watches with analog displays work the same. I don't know about Big Ben, but it's possible the author is wrong about that example.