That must be a regional thing Cee Gee. Nowhere that I've worked (East Coast, West Coast, and several locations between) has a notation of "found" in itself been used or taken to be a value statement about the pedigree of the monument. It simply means that it was in existence prior to the survey which is the subject of the map having been performed. Additional notations or comments are needed to speak to the pedigree, value or lack thereof for any particular found point. The notation could be "found 1" IP w/ tag, 'LS 0000' per RS 7/53". That more complete note states that it was a pre-existing point and states a pedigree. Showing it either as marking a particular corner or point, or being off from that corner or point as you've determined it will demonstrate your estimation of the monument's value.
In your example, you found a pin that you identified as one of yours, and you identified it as found. So you and I apparently agree that one can find their own pin. Where we appear to diverge is that you identify it as "found" only if it has been moved from the position you set it in, otherwise it is "set". I am of the mind that one does not set it during the current survey, they have "found" it, and it is either found (hopefully) right where they left it, or it is found where someone else moved it to. But this is a semantic argument.
IMO, the important part of identifying the monument in terms of pedigree is that it is noted to have been associated with some earlier survey. This can either be noted as "found 1" IP w/ tag, 'LS 0000' per RS 7/53" (assume you are LS 0000 and RS 7/53 is your filed map), or as "1" IP w/tag, 'LS 0000', set by this office in 1987 as shown on RS 7/53". Personally, I prefer the latter. Showing a preexisting monument simply as "set" without reference to the previous survey gives the incorrect perception that it was set during the current survey, possibly diminishing it's pedigree.
[Edit: Cee Gee, I realize that you are not in a recording state, so substitute the appropriate unrecorded map/survey reference for "RS 7/53", which refers to a recording book and page]
The only people looking at the map and caring at all about the license numbers on the tags or caps will be another surveyor who later uses your map and monuments in conducting another survey. A surveyor should notice that the license #s on those found points are the same as that of the surveyor signing the more recent map,
Don, that seems to me to be a pretty rigid and rather emotional response for what is more of a semantic or style issue than it is a substantive one.