= Solution
The <instantaneous reproduction number> at time $t$ is
$$
R_t^{\mathrm{inst}}=\sum_{\tau\geq1}\beta_{t,\tau}.
$$
It is the expected number of secondary infections that one infected individual would produce if the transmission conditions at time $t$ applied throughout that individual's infectious life. The <case reproduction number>, also called the effective reproduction number in this question, is
$$
R_t^{\mathrm{case}}=\sum_{\tau\geq1}\beta_{t+\tau,\tau},
$$
the expected number actually produced by a person infected at $t$ as calendar-time conditions subsequently change. The instantaneous quantity is easier to estimate in real time because it depends on current and past <incidence>; the case quantity depends on future conditions.
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