Effect of position on a motorbike turning While searching tips for driving a motorbike on wet roads safely, I heard some pilots saying to lean inside the slope makes the bike less tilted, so it's safer.
I agree leaning make the bike less tilted, but the overall "motorbike+pilot" object isn't less tilted. The bike is less tilted because the pilot is more.
Here my drawing of the two situations :

a. is normal position
b. is tilted position
green crosses are centers of gravity of 'motorbike+pilot'
For a. and b. :
 - same speed
 - same mass, so same weight
 - same angle from blue line (line from contact point to the canter of gravity of 'motorbike/pilot') to the floor
 - same centripetal force
 - vectors (in red) are the same  
On my bike (Yamaha TW) tires are perfectly rounded, so the surface in contact with the floor is the same whether or not it's tilted. So I don't see a difference.
Am I wrong somewhere ?
How can it be safer ?
 A: It is mainly about the so-called "effective steering angle" which is a function of the tilt angle and not necessarily equal to the steering angle measured at the handle bars. 
The sensibility of the tilt angle with regard to the steering is much higher at high tilt angles. This is because the sensibilty of the effective steering angle with regard to the steering is higher at high tilt angles. This means that the bike is easier to control, or less sensible, at lower tilt angles. 
Although the total center of mass is in both pictures tilted equally, in the picture b) the tilt angle of the bike is lower and thus the rider can control the bike by steering more easily.
A: First of all a side note:
The angle of the blue lines isn't the same, as the tires have a certain cross section and shape. As soon as the bike is tilted, the point where it touches the ground moves out of the center line, into the tilt direction.
In an extreme case, the bike could stay vertically, and only the driver tilts sidewards.
However, I see three reasons, why b) is safer:
While a) feels / is more stable as bike and driver are one unit, it needs more force to erect both into a vertical position. This force is not applied by the driver himself, as he can't erect the bike. Instead, he steers the front wheel deeper into the curve, which causes the whole unit to erect. At that moment, the force onto the ground increases, and the horizontal component may exceed the static friction.
In case of b), the bike is in a more vertical position and the driver can erect on his own. This causes less force onto the ground.
For almost the same reason, the driver can react faster when he sees that the condition of the street ahead changes, or may be, when the wheels already start to loose friction (which needs high skills).
As driver, you may imagine driving a curve very slow, once in the 'normal position', and once 'tilted'. Imagine you manage to stop within 0.5m, because a ball is rolling on the street or similar. It's quite impossible to erect from the 'normal' into the vertical position, while it shouldn't be a problem from the 'tilted' position.
Another reason is additional torque on the tires. The surface of the tire  which touches the ground has the size of... may be an egg. In a curve, the inner and outer side of this surface have a slightly different speed over ground, while the speed of an upright wheel is the same. This difference has to be compensated by the rubber of the tire, and creates a torque around the vertical axis.
For a certain angle of the wheel, this torque can be canceled out, as the inner side of the surface touches a point of the tire, where the wheel radius is lower. However, this usually happens at a fairly small angle, and with increasing angle, torque increases into the opposite direction. And of course, this torque should not exceed the static friction.
So, even in this case, b) is safer, as this torque is lower.
A: The Physics of riding a motor bike even in a straight line is complex so I can only make a few suggestions.
The frictional force on the tyres due to the road helps to get a motorbike around a corner.  
The Yamaha TW 200 looks like a motorbike which can be ridden off and on road.  If the condition of the road/track is  variable then the more upright the motorbike the more control one has when there is a sudden change in the amount friction.
If the bike and rider were a point mass then $F=mr\omega^2$ which will mean that for a give $\omega$ a smaller radius requires a smaller centripetal force.
The right hand picture moves the centre of mass such that the radius is smaller and so less frictional force is required?
This reducing the radius can be achieved by leaning the motorbike and rider over into the bend and a fairly recent technique developed by those who race motorbike is to actually move their backside in towards the corner.
When doing a slow turn to the left the advice is, move your backside to the right.
When moving slowly the bike is nearly upright and moving your mass in one direction moves the bike in the opposite direction and so it leans into the corner.
At higher speeds the technique is to lean into the corner as per the right hand diagram above but at the same time push on the handlebar on the same side as the corner - this is called counter steering.  The tendency for the front wheel is to turn into the corner any you have to counteract that.
Coming out of a corner to make the motorbike lean less a bit of acceleration is used.
I am not sure about @PierreKopaczewski 's tyres but modern tyres as well as having a rounded profile often have dual compound tyres.  The tyre compound in contact with the road when the motorbike is vertical is harder and hence more durable than the compound which is in contact with the road when it is leaning over.  The softer compound affords more grip but there is increased wear. On the other hand for a lot of riders a motorbike usually spends more of its time in the near vertical position.
