Gonzalo-
In general it would be good for you to state specifically which code you are working with. Based on your UG-10 reference I will presume that you are working to ASME VIII Div. 1.
Sounds as though the vessel has already been fabricated. It's a bit late to think about an ASME code stamp.
If the vessel was not yet out of the fabrication shop, you could compare the MTR to see whether the material meets some SA grade.
My understanding of API 5L is that it is quite a bit looser than ASME vessel materials. Just look at the huge range of tensile stresses:
Specified minimum yield could be as low as 25,400 psi or as high as 120,400 psi while specified minimum tensile values vary from 45,000 psi to 132,700 psi.
Needless to say, the chemistry and steelmaking practice for 5L has a wide variation.
Most of the steels used for Section VIII work have a yield : tensile ratio on the order of 70%. This allows for a bit of plastic action before hitting yield, effectively giving some forgiveness of local strain issues. On the other hand, your 5L X70 has a yield of 70.3 ksi with an ultimate of 82.7 ksi for a yield : tensile ratio of 85%. This is a substantial reduction.
The vessel is probably usable, but be careful of details which may be in the 1.5*S or 3*S range. But you probably cannot make it Section VIII compliant and put a U stamp on it.