If the older lathe has plain (babbit) bearings, there will be a certain amount of deflection always whn the lathe is at rest. When the lathe is turning, the oil film acts as the bearing and runout should be negligible. If the lathe has roller bearings, the amount of runout is related to the precision of the bearings and the preload on the beaings. Tapered rollers should be adjusted so there is a certain amount of preload (often this is established by the use of a Bellville washer under the adjustment nuts). Roller bearings are ajusted so there is just no play with very little loading.
Chucking a bar and pushing on it to check deflection is a viable technique with a couple of caveats. First, be sure and mount the base on the headstock and not on the bed or carriage. You want to check the bearings; not the rigidity of the machine. It is often a good idea to apply the pressure vertically to check for bearing slop since the spindle will naturally rest at the bottom of the bearing. You can even just indicate on the chuck or face plate.
On a typical toolroom lathe, it is relatively easy to deflect the chuck a certain amount because you can actually flex the headstock.
On my own 13x40 lathe, I can see about .0003" (three ten thousandths) of deflection if I lift up as hard as I can on the chuck. I can curl a hundred pounds (not too many times, mind you!) so I would guess I 'm lifting about that hard on the chuck. If I pry on the bottom of the chuck with a bar, I can, of course move it a bit more but not a lot. If the bearings were loose, the deflection would be greater and would be accompanied by a clunk as the spindle was bumped up and down.
As I said, plain bearings are a bit deceptive. A plain bearing must have some clearance (usually a minimum of .00075 " radially) to allow for lubrication and the establishment of a dynamic oil film around the shaft.
If you clamp a cylindrical piece in the chuck and dial it in to within .001 or less TIR (total indicator runout), dialing it at the other end is valid only if the cylindrical portion is co-axial with the bore and the chuck jaws are perfectly aligned with the bore of the machine. If you are to dial both ends to run true, you have to provide a means for the end being held in the chuck to swivel. If the barrel is cylindrical, this can be accomplished by using a piece of 1/8 inch brass rod or wire between the chuch and barrel. If the barrel is tapered, it will swivel just fine.
At the outboard end of the spindle you must have a means of centering the barrel. A spider with four screws to adjust the barrel to center works well. The installation of four screws in the outboard end of the spindle itself also works well.
You can also crown by setting the barrel up in the steady rest. If the lathe lacks a sufficiently large bore to allow the barrel to go through the headstock, this is the only way you can crown it. You can use an adjustable spider on the barrel to center it up in the steady (the breech end is held in and driven by the chuck) or you can install a cat's head on the barrel. This is a sleeve over the barrel which is then turned with the barrel supported on the tailstock center so that the cat's head is concentric to the bore. The steady rest is then set up to so the cat's head runs on the steady bearings. Some run the stready directly on the barrel but there is likely to be some runout if it is done this way.
With more experience, you will learn different set-ups and methods to minimize error. You will also find that, with experience, instruction which seemed like gibberish will suddenly make sense. Good luck! GD