To add to the answer from Bioathlete, "reads" in the traditional genomic sense are the string of nucleotide data inferred from the sequencer from a given fragment of genetic material (could be Illumina, Nanopore, PacBio, 10X, etc.).
So read count first applies for all of the "reads" defined above generated by the sequencer. From a NovaSeq, the total number of bases from a single flowcell is up to 3 trillion bases at 150bp each read for the highest throughput flowcell (so a read count of 3,000,000,000,000/150 = 20B total read count).
The read count for a single Nanopore flowcell is roughly (30 billion bases/avg read length), but the read lengths can be SUPER long (100kb compared to 150bp is substantial). Long read technologies are generally used for different analyses than short reads.
So all of the above reads would be defined as "raw reads", meaning they are reads not yet analysed in any way to be used for data analysis. So using illumina as an example, for a single Whole Genome sequencing sample we might have 800,000,000 raw reads. We then use an aligner like bwa mem to map the reads to a single reference genome (or in your case you would use a different aligner to map to multiple genomes). From here we still have the concept of raw reads, but we also now have the total number of mapped reads. A typical good illumina library for a single reference genome will have >90% mapping rate for a human sample.
We can then say we have 800,000,000 raw reads, and 720,000,000 mapped reads with a mapping to a reference genome of 90%. So we can compare "raw read count" to "mapped read count" to get an idea of the quality of our library.
In the case of metagenomics, you would consider your "raw read count" to still be the total number of reads you got off the sequencer, but the "mapped read count" would be the sum of the reads mapped to each individual reference genome to get a total quantity of "mapped reads" which could give you an indication of library quality. I'm not sure what a good metagenomics library "mapped read count" is, but I'm sure you can find that information.
Lastly, we want to get an idea of the depth of coverage for each mapped read to the genome. This will tell us how well our mapped reads actually covered the reference genome or genomes we are interested in.
I hope this was helpful!