Since the TCR is a value given for a particular wire, and since I do not see a range being given, I will assume that TCR is not variable and does not need set for a particular temp range. Setting the temp on the SX Mini M with the current firmware is to allow the device to know the starting temp of the unit/coil so it can adjust its internal algorithms away from the default 68F value.
This is the info from the Wire Wizard for Ti (Grade 1):
TCR in vaping range 3660 ×10-6
The info box that appears when you hover over this is: The relative change in resistance between 20C (68F) and 300C (572F)
Note the fixed number for TCR for a given wire. The given TCR does not change with temp.
Fine, you are coming to the point. Steam engine declares the TCR it displays as the coefficient that linearly interpolates the two R-T points corresponding to T0=20°C (68F) and T1=300°C (572F).
If Lars Simonsen (the author) had chosen to interpolate the R-T curve at T1=427°C (800 F), the TCR would be 0.0035, exactly the TCR value reported in your first post and this is not a coincidence.
If you chose titanium 1 in the wire wizard and click at the graph tab in the Temperature control results box, you will see the real R/R0 vs T curve where R0 is the resistance at T0 ( R/R0 is known as the Temperature Factor of Resistance or TFR). Actually also the TFR curve is an approximate representation of the real material behavior (it is a piecewise linear approximation of experimental data at temperature points [-50,20,100,150,200,250,300,427 ] °C) but this is not important here.
What is really important is that you realize that the R-T curve is not a perfectly straight line . Thus if you want to represent that curve with ther linear function
R=R0 * ( 1 + TCR * ( T - T0) ) [#]
you have to decide how to evaluate the TCR coefficient from the experimental data. The simple way is linear interpolation at a given temperature T1, that is by imposing
R1=R0 * ( 1 + TCR * ( T1 - T0) ),
that is
TCR= ( R1/R0 -1)/ ( T1 - T0) = ( TFR1 -1)/ ( T1 - T0)
Choosing T1=200°C (392F) , being TFR(200)=1.66318 (experimental data) you get TCR=0.00368
Choosing T1=300°C (572F), being TFR(300)=2.02477 (experimental data) you get TCR=0.00366
Choosing T1=427°C (800F), being TFR(427)=2.45116 (experimental data) you get TCR=0.00357
Lars decided to evaluate the TCR at T1=300°C (572F) because doing so the linear approximation [#] is exact at 300°C and very accurate in the whole 200-300°C range used in TC.
I hope I have clarified the issue.
Regards