My leafblower motor burnt out, trying to rewind the field winding - armature looks OK. So,

1. Is it easier to wind all the turns on a bobbin and then transfer to the motor pole or wind turn by turn in the motor. Winding on bobbin means I have to get the length of turn just right.
2 How to specify magnet wire i.e. just describe as magnet wire with gage to buy on amazon?

 

There are too many unknowns to answer your first question. A couple of pictures of the motor might help.
The number of turns and the gauge of the wire on the field winding are the most important things to consider. If you don't get them right, the performance of the motor will not be the same. The existing winding will be on an insulated former. That is what you should use.
To do the job properly, you should unwind the old winding, counting the turns that were on it. Then you can rewind it, using the same gauge of wire and the same number of turns. There will probably be layers of insulation between the windings. Save these, and re-use them in the same way.
The answer to your second question is yes. The wire is enameled copper wire or magnet wire. It is sized by gauge (AWG) or by diameter. Make sure you get enough to do the job. You don't want to have a join in the middle of the winding.

 

The way we used to do it when I did a spell at motor winding, was to shear off one end of the stator coil with a cold chisel and count the turns the rest of the coil remaining could be usually driven out of the slot.
Winding is a little easier if a coil bobbin is made to the right size and then use the bobbin in a lathe, or similar way of revolving the bobbin in order to wind the coil on.
As stated , try and retain the insulation material in order to reuse..

 

THANKS. Here is a pic of the motor. Difficult to remove the old wiring. Also, the only source for mag wire is Amazon?

 

FORGOT the pic!

 

Now that I have seen the picture, if it was mine, I would not even try to fix it.

 

Buy a new one.

The new Wire required for rewinding may be more expensive than a new Leaf-Blower,
and You can make ZERO mistakes,
this is not a simple job,
and an Electric-Leaf-Blower is a Home-Owner-Gimmick at best.
It is not designed to last for years, so even after a perfect rewind, it's guaranteed to fail again.
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