The definition of the thermal coefficient of resistance (TCR) is the change in resistance per change in temperature divided by the resistance at a specified, fixed reference temperature:
$$ \mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}}. $$
Note that $R(T_\mathrm{ref})$ is a fixed value. It is the resistance given at a single temperature $T_\mathrm{ref}$. It does not depend on $T$. Typically $T_\mathrm{ref}$ is specified to be $20^\circ\mathrm{C}$ or $25^\circ\mathrm{C}$ ~ room temperature. Less often $0^\circ\mathrm{C}$ is used for $T_\mathrm{ref}$.
With the above definition the TCR is a constant and the resistance can be written as a linear function of temperature around $T_{ref}$:
\begin{gather}
\mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \frac{R(T)-R(T_\mathrm{ref})}{(T-T_\mathrm{ref})}; \\
R(T) = R(T_\mathrm{ref}) \big(1 + \mathrm{TCR} (T-T_\mathrm{ref})\big).
\end{gather}
The TCR is going to be different for different reference temperatures. You have to provide the value used for $T_\mathrm{ref}$ along with the TCR! For example, suppose that I measure a resistor and find that its resistance is a linear function of temperature over a given temperature range (intercept $b$ and slope $m$):
$$ R(T) = b + mT. $$
Then the TCR for this resistor can be found using the definition of $\mathrm{TCR}$ and a given $T_\mathrm{ref}$:
\begin{gather}
\left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}} = m; \\
R(T_\mathrm{ref}) = b + m T_\mathrm{ref}; \\
\mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}} = \frac{m}{b+mT_\mathrm{ref}}.
\end{gather}
Note that the TCR depends on the $T_\mathrm{ref}$ you specify. The TCR will be lower if you choose a larger $T_\mathrm{ref}$. The $T_\mathrm{ref}$ should be specified along with values given in material property tables.
In the end, the value of a TCR is that it allows you to estimate the resistance of a resistor as a function of temperature given only a measure of its resistance at $T_{ref}$.
One more thing. If you know the value $\mathrm{TCR}_1$ at reference temperature $T_{\mathrm{ref},1}$, then you can find $\mathrm{TCR}_2$ at a different $T_{\mathrm{ref},2}$ using this equation:
$$ \mathrm{TCR}_2 = \frac{\mathrm{TCR}_1}{1+\mathrm{TCR}_1(T_{\mathrm{ref},2}-T_{\mathrm{ref},1})}. $$
