Does this mean I do not have different day and night rates?
If your bill shows only two sets of entries - primary & secondary units - then no, you are not on a scheme which gives different day and night rates. That's consistent with the tariffs you quote too. There are variations in prices from one supplier to another, but at the moment cheap night-rate power is generally about half your secondary units cost (e.g. my e-on plan currently charges 27.8p primary day, 11.9p secondary day, and 5.4p night per unit, exc. VAT).
The primary/secondary units scheme has been adopted by many suppliers in recent years in lieu of the older scheme which charged only a single, basic rate per unit but added a standing service charge of a few pounds per quarter.
The original intent of cheap night-rate electricity was mostly for electric storage heating, where the thermal bricks in the heater are warmed up using cheap power overnight, then release their heat gradually during the day. Some of the older plans had cheap-rate power
only for the heating, and everything else in the house was charged at the same rates day and night. These were commonly referred to as "White Meter" plans, since there was a completely separate meter used to record the energy used for the heating and this meter was white to distinguish it from the black meters used for everything else at the time.
These days, the entire house is switched to the cheap rate power overnight. There can still be a switching arrangement to turn on power for storage heating; the "clunk" you hear when the cheap rate starts is the contactor which applies power to the heaters operating.
Economy 7 is the most common plan, so named because it provides cheap rate power for 7 hours each night, typically midnight to 7 a.m. in winter, 1 a.m. to 8 a.m. in summer. In some areas the times can be a half-hour or so different though, and sometimes even along the same street you will find staggered times, since the timings are set up to avoid everybody's heating all coming all simultaneously and suddenly throwing a heavy load on the grid. Economy 7 gives you the cheap rate at night, but often with the penalty of having to pay slightly more per unit during the day (or a higher quarterly standing charge under the older schemes).
There are some other plans as well which can provide cheaper rates at certain times, for example Economy 10 which is basically Economy 7 plus some mid-rate power for 3 hours during the afternoon.
we're not sure which one is right.
The meter usually indicates which rate is in operation at any given time. On a newer meter with LCD readout, it will have something showing "Rate 1" or "Rate 2" on the display which will change at the appropriate time (there are many different types of meter in use, so it's impossible to be precise without knowing which sort you have). On the older type of meter with a mechanical readout, there will be a little mechanical flag of some sort, typically an arrow which rotates to indicate "Normal" and "Low" (or "Day" and "Night").