The story is not clear about it, but apparently it was Claudius Ptolemy himself. He realized that if the day began at dawn, the accounts no longer matched. So the Romans began using midday (when the Sun is higher) as a fixed reference. In fact, they come from the terms "a. m." (ante meridien, or before noon) and "p.m." (post meridien, after noon).
The opposite time was midnight. The custom extended from the Romans to the Christians and from there to the whole world. Source of the Image

The common people really did not care about the minutes or the seconds, just knowing more or less in what part of the day they were. It was several centuries later that these divisions were included as fractions of an hour in mechanical clocks.

Fun fact: for the Hebrews the day begins at nightfall.

The precision of these devices only became important until middle age, when it was necessary to keep a strict schedule of prayer times and jobs. Although we continue to use a system based on customs from millennia ago, technology has been fundamental in keeping our time measurement up-to-date.

Today we almost do not use solar, sand or pendulum clocks, but we rely on more precise mechanisms such as atomic time. Since the 1960s, a second has been defined not as a fraction of an astronomical event but as the duration that it takes for a Cesium atom (Cs) to perform more than 9 trillion oscillations.

The accuracy achieved with this type of atomic clock is so high that it admits only one error of one second in 30,000 years.

The most precise clock in the world is designed at the Paris Observatory, where the current atomic clocks would take 52 million years to be out of phase for a second.

An atomic clock is so precise that we will never see a delay in our lives or in our solar system.

Atomic clocks will never reach our wrists.

The only problem with atomic clocks is that they require very stable environments and very cold temperatures for optimum precision. In addition to that the accuracy we have achieved with current watches makes an atomic clock for all people is not a thing with a large market or prospects of mass selling. However, we constantly see that there are a lot of people in the world who do not know what to do with their money.

Perhaps these people are willing to pay the exorbitant price that could cost a precise atomic wristwatch, simply to be able to say that they have something "unique", for the rest of the mortals the atomic clock will forever be a technological conquest as a species and an instrument to know more and to know better how is the universe in which we live.

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Reference.

1, 2, 3; 4, 5, 6, 7.