How to alert a bus driver to stop without pushing the stop button

In the meaning of the sign, asking him to make a stop is not a distraction. The sign is referring to irrelevant talk, particularly that requiring thinking: means don't ask for the time, ask his opinion on Brexit, union organize, etc.

Think of it like "Sterile Cockpit" for pilots, or what US NTSB says about focus on the driving task. Asking if the Dreamliner is fun to fly is not appropriate in sterile cockpit, but very correct to ask about a flaps setting that looks funny. Listening to a roiling political show is a distraction, but Siri saying "turn left 500 feet" is on-task.

Calling for a stop is definitely acceptable, as it is on-task and brief.

You can also say "I need Bristol Road".

If you can't easily reach a stop button and can't get somebody else to press one for you, it's perfectly reasonable to verbally ask the driver to stop. Indeed, when I was a child, thirty years ago, it was completely normal to just walk to the front of the bus and say "Next stop, please" rather than pressing the button. That always felt more polite than pressing the button. Today, everyone seems to just press the button.

However, if you have to raise your voice to get this request heard, it's almost certainly inappropriate. Yelling at the driver will certainly distract them from driving and their first thought will be that something is seriously wrong. You shouldn't do that. Worst case is that you'll miss your stop but stops are usually pretty frequent so having to walk back a stop isn't going to be the end of the world.

It would be better to be more prepared. As you get within a couple of stops of where you want to alight, you should be able to manoeuvre yourself within reach of a button. As people get off at the stops before yours, people are moving around the bus anyway, which gives you an opportunity.

Is this “distracting”?

YES. Heavens, yes! The driver is alerted primarily by the audible chime so as to not distract their eyes.

Though they're likely trained to ignore passengers, there's still an internal push to focus on a person demanding attention.

What other ways could I alert the drive or do I need to press the Stop button?

You can pull the cord if the bus is also equipped with one.

If the bus is very crowded, hollering "Next Stop" well outside the drivers field of vision isn't the worst thing in the world, but still, push the button or pull the cord.

The reason I am asking is because at the top near the driver it says not to obstruct or distract him. […] Is this “distracting”?

This is a bit quite funny confusion.

No, it isn’t. A sign asking to not distract a bus driver means you should not try to discuss with a driver a recent football game nor discuss it loudly by phone with your friend while sitting near the driver and not to obstruct means avoid putting your snowboards in a way that reduces driver’s visibility.

A stop (bell) button is actually designed to avoid engaging a (possibly distracting) conversation with a bus driver just to say a bus should stop on the next bus stop.

Drivers expect that signal and it is less distracting than trying to ask a driver politely in a few words.

Please note

Also, remember to ring the stop button (or pull the cord) slightly before approaching the bus stop as the bus driver need some time to respond and slow down a bus safely, if you call the button when a bus is already passing a bus top it might be too late for a safe stop and a driver might skip it (not being happy about your late response.)

P.S.

Probably a bus operating company should update that sign to be more clear and avoid such confusions by adding something likeL “press a stop button when approaching a bus stop you want to exit.”