Round is the shape.
Brilliant is how the gem is cut.
The modern round brilliant consists of 58 facets
Gemstones can be measured in different ways.
Three measurements: Length x Width x Height (depth) for oval or polygon gems
Two measurements: Width x Height (depth) for round gems
Sometimes only one measurement is given: Width. i.e. 5 mm round / square gemstone
Round gemstones should be symmetrical with a perfectly circular outline. However, in practice, a good percentage of gemstones are native-cut without the use of precision lasers and other advanced lapidary tools, so many round gems are not actually perfect circles and are rather better described as being near-round. This is why you may find that many still use three measurements for round gems. gemselect.com (not an endorsement of their store or products. Information only)
Bottom line: To get something the same size as the stones you have you will need to measure them and then look for stones with the same numbers.
Mass produced jewelry settings come in standard gem sizes. Some makers will cater to these sizes and make stones that fit. If the stones you have are a standard size you increase the chances of finding matches.
Note: the linked standard gem sizes chart shows a size by weight. The chart is for diamonds - other types of stones will have a different weight to size because they have different densities.
Even if you find something with the exact same numbers different makers will have different tolerances and have different levels of consistency. You will have to decide how close is close enough.