There are two angles which are important in determining the performance of a lathe tool; rake and clearance.
The rake angle is the angle between the leading surface of the cutting edge of the tool and a plane through the axis of rotation of the part and the point of contact of the tool. If the leading surface slopes away from the cutting edge, it is a positive rake. A tool can have either back rake, side rake or both.
The clearance angle is the angle between a plane tangent to the part surface at the point of contact of the tool and the trailing surface of the tool. A tool can have either back clearance, side clearance or both.
The drawing shows a part being turned with a tool coming in on center (blue), above center (green), and below center (red). The tool is presented horizontally and when on center has zero rake. If the tool is moved to an above center position, the rake becomes positive but the clearance angle is such that there is interference with the work, causing rubbing and a tool the won't cut. If the tool is moved to a below center position the clearance angle has increased but the rake has become negative
Different materials work best within a range of rake and clearance. Here is a site that lists those angles.
http://www.steves-workshop.co.uk/tips/toolgrinding/toolgrinding.htm As can be seen, moving a tool up or down from the center line can have a detrimental effect on cutting performance. By grinding a larger rake and clearance angle, the tool becomes less sensitive to vertical position but the strength and durability of the tool is decreased. Furthermore, if you are using insert tooling, you have to work with the geometry purchased.
There are two more reasons for running a tool on center.
First, when off center the angular relationship changes with the part diameter so as a part is being turned down, the rake and clearance are continually changing. In fact, as you are approaching center there will be a point where you are shearing off the remaining nub or riding under it.
Second, machining off the center line distorts the geometry of the cut. The tool moves in a horizontal plane but material is removed radially and the radial motion is the horizontal distance times the cosine of the angle formed by the plane through the centerline and the point of contact and the horizontal plane. And that angle is continually increasing as you get closer to the centerline.
It should be noted that in certain circumstances, it may be desirable to set the tool above or below the centerline as long as proper rake and clearances are observed. There is a complicated mechanical coupling between the work and the cutting tool that will vary greatly from one machine to another. This greatly affects chatter and surface finish. Each setup will have its own sweet spot.
So how accurate do you have to be? A 3º angle has a cosine of .9986 and a sine of .052. On a 1" diameter cut, that would amount to a vertical offset of .026" and the cosine error would be negligible for most work. I use the vertical ruler at times. I also use a gage like this.
https://www.ebay.com/itm/Lathe-Gage...hash=item5649056a5f:m:m3NZb7KTSDbcKWZzGIrJgqA
There are a number of DIY tool height gages which rest on the cross slide. A final check is what happens as the cut approaches the center of the part. If you are low, there will be a nub left or it will break off as you move past center. If you are over, you will shear the nub off.
Edited to include missing drawing