As per my knowledge, the number 18 refers to the core/main/important layers the architecture is based on i.e., the convolution and fullyconnected layers. In vgg16 it would be 16 and in vgg19 it would be 19 etc.
You can use analyzeNetworkfunction to analyze the network architecture. So if you check the count of only the convolution and fullyconnected layers in the resnet18 pretrained network, it will be 21, but in these the layers with names "res3a_branch1", "res4a_branch1" and "res5a_branch1" are 1x1 convolution layers which are used to control the depth so that the skip connections are possible accross the branches. Hence removing these 3 would leave us with 18. You can check the same with other variants of resnet as well (resnet50, resnet101 etc).