Please explain this formula for the relationship between bandwidth and the temporal response of a photodetector
The equations shown are all derived from the same formula which you already gave:

$$t_r = \frac{0.35}{B}$$

A single-pole system has the frequency response

$$H(s) = \frac{A}{1 + \tau\cdot s}$$

This sytem has a pole at

$$p_d = \frac{1}{\tau} \Rightarrow BW = \frac{1}{2\pi\cdot \tau}$$

The transient step response of this system can be calculated as

$$h_{out} = A\left(1 - e^{-\frac{t}{\tau}}\right)$$

From which you can calculate a timepoint for each output value:

$$t = -\tau\cdot\ln\left(1 - \frac{h_{out}}{A}\right)$$

If you'd rather use the rise/fall time instead of \$\tau\$, you can then easily calculate that

$$\tau_r = -\tau\cdot\ln(1-0.9) + \tau\cdot\ln(1-0.1) = \tau \ln(9) \approx 2.1972\cdot \tau$$

Hence

$$BW = \frac{1}{2\pi\cdot \tau} = \frac{\ln(9)}{2\pi\cdot \tau_r} = \frac{\ln(9)}{2\pi\cdot \tau_f}$$

The abbreviations are the ones used in French. "Bande Passante" (BP) means bandwidth, "Monter" (\$\tau_m\$) means to rise, "Descendre" (\$\tau_d\$) means to fall.