What should the transfer function be for this RCL circuit?
A filter necessarily processes some sort of signal, so the transfer function that makes the most sense is the one that describes the filter's processing of the signal of interest. If the input and output signals are both voltages (e.g. the filter input is from, say, a voltage amplifier, and the filter output serves as the input to a voltage buffer or amplifier) then the transfer function of interest is
$$H(s) = \frac{E_{\text{out}}(s)}{E_{\text{in}}(s)}$$
The reason is that you will presumably want to determine the transfer function of the entire signal chain, in which case you can most easily combine the filter's transfer function with the transfer functions of the input and output stage(s) if you've determined the filter's transfer function in the form that matches the input and output stages' transfer functions.
Many filters operate on voltage signals so in the absence of any other information about the signals (e.g. you are analyzing the filter in isolation) I would use this transfer function. Note that loads at the input and output will often affect the transfer function regardless of the signal of interest -- e.g. a finite load impedance at the filter's output will reduce \$E_{\text{out}}(s)\$ compared to an infinite load impedance -- so it may not be particularly helpful to analyze the transfer function of a circuit in isolation anyway.
If the input and output signals are both currents then the transfer function
$$H(s) = \frac{I_{\text{out}}(s)}{I_{\text{in}}(s)}$$
would be more useful.
While the signal of interest is often in the same form at both the input and output (e.g. both input and output are voltages) it is possible for the input to be of a different form than the output. For example, if the filter input is connected to the output of a voltage amplifier and the filter output is connected to the input of a transimpedance amplifier (which uses current as its input signal) then the transfer function
$$H(s) = \frac{I_{\text{out}}(s)}{E_{\text{in}}(s)}$$
would make the most sense.
The transfer function
$$H(s) = \frac{Q_{\text{out}}(s)}{E_{\text{in}}(s)}$$
would be useful if the filter input was a voltage signal and the filter output was a charge signal, but a charge \$Q(s)\$ is not a common signal (at least not directly -- perhaps the filter is charging a capacitor, but in that case I'd still expect \$I(s)\$ to be considered the output &quot;signal&quot;).
All this applies to any filter (not just a bandpass) or, for that matter, any circuit block: the most useful transfer function is that which matches the input and output signals of interest.