Today we just say the time since we tend to use digital clocks. The cited rules from the Cambridge dictionary are more historical, likely owing to how analog clocks are read differently than increasingly common digital clocks.
As noted in the comments, the transition from analog to digital clock faces hasn't been completed, and many folks still see analog clocks as the norm. This likely varies with culture and within cultures by generation and setting. It's likely that the choice of wording also varies along the same lines as a consequence.
The style of expression given in the Cambridge guide has been going out-of-fashion, though is still retained by some speakers. This appears to be a technology thing.
In olden days, clocks used to look like this:
When you're looking at a clock like that, it makes more sense to regard minutes until the next hour.
But, as time goes on, digital clocks are becoming more prevalent:
Given this display, it's kinda weird to restate the time in terms of the distance until the next hour.
So, today, it's increasingly common to just read the time.
Time-telling precision has changed, too
Historically, it didn't make sense to give the exact minute for two big reasons:
It's often difficult to tell the exact minute an analog clock intends to represent without getting a clear look at it, which often wasn't worth the effort.
Historically, clocks had to be manually set and the clock's displayed time would drift until updated again. But even when recently set, the setter would tend to have another imperfectly set clock providing the set time, causing further errors.
Given these factors, it didn't make much sense for folks to work to distinguish the exact time.
Today, digital clock faces provide the exact minute without any additional effort on the viewer's part, and many digital clocks, e.g. those on computers and cell phones, auto-update their time to a central time authority. This means that it's now easy and meaningful to specify a minute, causing a reduced interest in approximation rules.
A commenter has noted that they still use analog clocks, and that they still use the corresponding language. So, I guess, it's probably more precise to say that the terminology correlates to the type of clock that one is used to; folks who still use analog clocks are more likely to use the prior terminology.
I've referred to analog clocks as "historical", though it's likely that many adults are old enough to remember when analog clocks were the norm. I'd speculate that there's likely a generational difference on this issue.
For English learners, it's probably worth pointing out that the style of English I employ on StackExchange is intentionally informal. Here are a few things to compare/contrast:
I used "kinda" above, which is a slur for "kind of".
I used "olden" above, which is an informal term often meant in a humorous sense. Part of the joke is that the term "olden" is itself archaic, so it's self-referential humor.
I used "don't really" above. Use of "really" in this context is typically regarded as informal, and excessive usage of it can come off as juvenile.
I used "technology thing" above. Describing something as an "X thing" is typically informal.
I expressed a disregard for a style manual, specifically Cambridge's. This bluntness can come off as irreverent, as was the intent above. Part of the subtext there was that English is about communication; grammatical rules are things to know and understand, but not to simply conform to as laws.
New learners are typically better advised to stick with standard grammatical rules. Once mastered, stylistic alterations can be adopted.
The above answer is about the dynamics of how English is evolving as a consequence of the shifting experiences of its speakers. Personally, I find this to be fascinating — and, better, it's something that can be studied empirically if data's found.