Choice of primers for PCR
You do need two pairs of primers one unique to each species as you need a positive result to be confident that the thing you are looking for (cow or horse) is there.

There is no limitation on them being in the same place (with a different last base).

Usually this would be analyised by gel-electrophoresis and not by sequencing so you don't have to worry about getting the whole thing. This is because you are only looking for the presence or absence of PCR product.

When you do the PCR you will have 4 reactions, one which amplifies horse only, one which amplifies cow only and their negative controls (adding water instead of lasagna DNA extract). After the PCR you take a small amount of your PCR product and run it on a gel (gel-electrophoresis). If you see amplification in the negative controls then you have contamination and can't trust the result. Otherwise if you see amplification by the cow primers then cow is present in the lasagna and if you see amplification by the horse primers then you probably don't want to eat the lasagna. If you used both options for the first forward primer you suggested (blue) then all your bands would be the same size, if you used both the first (blue) and second (yellow) then you would get different band positions on the gel due to the different sized products which makes you feel more sure that you didn't make the mistake of adding the same primer pair to both tubes (it happens).

In the end it's simply a matter of style.