Well you answer your own question; a rich man could walk down the street at any time of the night. Where would a rich man walk? The answer to your question lies there-in. London was certainly one of the most policed cities in the world at this time, but how it was being policed differs greatly from how most people now would imagine Police should operate. Also you have to define what is 'safe' - is a city 'safe' when it has a lower absolute crime rate, when it has less organized crime, or when its lacking simply in violent crime? Should we discount 'property crime' as a threat to public safety?

Policing strategies at this time had very much not moved past rudimentary presence and basic patrols along well-lit public areas. Focus was on solving crimes once they were committed and reported, not on true prevention. This meant that bobbies wouldn't go into the back alleys and talk to the disenfranchised unless they truly had to - such as when there was a reported crime.

I, personally, feel the larger the city the more dangerous, the more crime

Not necessarily! The larger the city, potentially the larger and more well-funded the police force. Its far better to look at the potential social gaps in a city to establish what its potential real crime rate is. Are there social enclaves? If so, are these social enclaves well entrenched? Such formations of social enclaves, especially if met with government indifference (e.g: A lack of support, either through the legitimate inability to support and monitor or the lack of motivation) can create breeding grounds for social deviancy and possibly criminal deviancy, or worse, organized crime.

Does London fit this description in 1910? Absolutely. Even as far as 1910 the effects of industrialization are still visible. Yes, unions are being formed and regulations are becoming more stringent at this time, but the reality is that there are many people living in similar conditions that are underpaid and very much disenfranchised.

So, there's no 'umbrella answer' to the question:

So, asking the experts, how safe was London, really, in 1910?

I have no doubt that the public, well lit areas where the Police (and the government that funded them) had the motivation and willingness to monitor and patrol actively were relatively safe. However, that could seem much a different world to those who lived in social enclaves, be they economic or ethnic enclaves where the Government neglected to show care or diligence to. I must stress, crime and social disenfranchisement is only co-relational, the true cause for the former leading to true crime is almost always government failure or indifference. Lack of funds, lack of support, lack of presence is what causes the disenfranchisement and encourages or indeed necessitates social deviancy.

The idea is that the victims of crime are most often drawn from the same group committing the crime, and the callous premise of 'out of sight, out of mind' very much dominated the strategies of policing in this broad time period. Its only of course heightened by the fact that social enclaves, by nature of their social disenfranchisement, were far less likely to report property crimes or often violent crimes.

Edit: In short, the police were there more or less to contain social enclaves, when elements from the social elements most likely to engage in criminality 'spilled over' into other areas, and the crime were likely therefore to be reported, police activity would then be highly visible.

Edit 2: /u/Bodark43 and /u/yupishi provide two very good ethnographic studies relevant to this.

Not sure about 1910, but if you're interested in 60 years earlier, Henry Mayhew's 1850 "London Labour and the London Poor" is very interesting and surprisingly readable, his aim was to tell the stories of the poor in their own words, but because the book was aimed at the educated classes, he had to make it readable to them. Result is very candid and interesting first-person accounts and Mayhew's discussions. A lot of the slang is very familiar!

A pretty good brief portrait of the place in 1903 is Jack London's People of the Abyss, when he spent some time posing as a homeless laborer. Some allowance has to be given for London's need for dramatic effect ( always beware of long verbatim quotes) but there is a pretty constant policing, mostly ( in London's view) to manage and restrain the huge numbers of desperately poor people, and he mentions little violent crime other than domestic abuse - no gang warfare, for example.