The statement that an active DI is a preamp is not correct. A passive DI has a limited input impedance, usually somewhere between 10k and 50k ohms. This is low for a passive instrument, and will load down the pickup, causing a loss of highs, complexity, clarity and probably level. A passive DI, placed between a passive bass and an amp can seriously affect the tone of the amp as well due to the loading.
An active DI will place an input buffer between the input and balancing transformer. It will usually have an input impedance of 1meg or higher...perhaps 5-10meg ohms. The buffer is usually unity gain or slightly less, but will pass a passive instrument transparently, maintaining the full frequency response of a passive instrument, and doesn't load down the signal if placed between the bass and amp.
If you have a low impedance out, active bass, or put the DI after a low impedance out pedal, the low impedance out will isolate the signal from the loading effects of the passive DI.
"Active" in a DI just creates a high input impedance and does not necessarily increase gain. It just protects against frequency and level loss.