If you have questions about anything in MEME Suite input forms or output, be sure to look for the red question mark icons near the text of interest. Clicking on these will bring up online help, which may clarify things for you.
Keep in mind is that MEME and STREME perform a very different function from FIMO. MEME and STREME are de novo motif discovery programs. Their goal is to determine if a sequence file *as a whole* provides evidence for the existence of a motif, and generates the position weight matrix (PWM) that characterizes any such motifs. FIMO is a simple motif match search program. That is, given a PWM, FIMO identifies every position in the sequence file that matches the PWM for the motif, filtered by a p-value or q-value threshold.
Although MEME and STREME are performing the same general task, they use very different algorithms, and mean very different things by “site counts”.
MEME requires that you specify a model of motif occurrences, “only once occurrence per sequence" (OOPS), “zero or one occurrence per sequence” (ZOOPS), and “any number of repetitions per sequence” (ANR). Given your choice of how motif sites are expected to occur, MEME will make some initial guesses about where motif occur and what the PWM for the motif is. MEME then iteratively refines those guesses until is can estimate a statistical confidence for the motif. It then stops looking for evidence for that motif, and moves on to looking for other motifs. For MEME, the reported site count is the number of putative motif sites that that were used as evidence for the motif. This count is by no means exhaustive, and in fact the minimum and maximum number of sites to be used can be chosen by the user under the “Advanced options” section.
STREME evaluates candidate motifs by looking for a single match to the candidate motif in each sequence. If a sequence contains a match it is labeled “positive”, and if it doesn’t it is labeled “negative”. There might be more than one match to the motif in the sequence, but they will be ignored. In that sense it is like the ZOOPS model in MEME. In the STREME output, the reported site count is the number of sequences in the target file which are marked “positive".
If you need an accurate counts of all matches to a motif I would not rely on the MEME site count number. If you are confident that each of your sequences will contain only one or zero instances of a motif, you could use the site number reported by STREME. However, if it is possible that your sequences could contain more then one biologically significant motif instance in a sequence you should scan your sequences using FIMO.
You don’t mention why you need motif site counts. If you are actually interested in comparing motif enrichment in target and control sequence collections, you may want to look into using other tools in the MEME Suite. SEA, Centrimo, or AME, are all specifically designed to look for motif enrichment.