A strain gauge will eliminate the problems. If you measure the crosshead displacement you will end up with too low numbers, but definitely not that low. Maybe your grips cause the problem.
Frank Kern strain values were not far away from the truth as both tests revealed that fracture strain is around 0.25 which is correct for steel ... the main issue is the linear zone and the strain value where material yields
besides, machine grippers, calibration, accuracy or any factor which may cause corruption in results was treated by conducting same specimen on two various machines and the results were almost identical
1) In order to measure E modulus you need strain gauge or extensometer. You could also use an image based extensometer if you have a good camera, make clear markings on the specimen, and synchronize video with data acquisition.
2) Converting displacement data to strain by simple division with length (which length?) will give you significant errors. If you want a relatively accurate estimation of strain you will need to estimate the machine compliance first based on an assumed modulus E.
In the elastic part of the curve you use the equation
u/F = C + length/(E*A)
to find the machine compliance C, based on a displacement u, a force F, the assumed modulus E, the gauge length "length", and the specimen cross section area A.
Then you estimate all of your strains (in elastic and plastic region) with
epsilon = (u - F*C)/length
This will also give you more accurate results for the fracture strain, which for your measurements will be around 0.23-0.25 instead of 0.27-0.28.