In the desadi tala compositions,the kriti begins 3/4 matra or 3 aksharas after the veechu. In the adi talam format, 1.5 beats after samam is right.
This is how I have learned deshAdi as well, veechu, samam, plus two finger counts with the actual eduppu at 3/4 into the veechu.
dEshAdi is not a meaningless thing as one may presume -- if you sing a kriti such as say brovabhArama with deshAdi you can see that the emphasis of the words in the kriti fall neatly on the samam of the deshAdi e.g. broVabharamaraGHUrama where the emphasis is on the capitalized syllables here and both these fall on the deshAdi samam. If on the other hand you put Adi tALam as we do it these days for this kriti, the emphasis on the "GHU" syllable, while it occurs in the singing, is totally lost in the tAlam. And this pattern will repeat all through the kriti. At the beginning of the anuppallavi "shrI vAsudEva" the emphasis in the singing falls on the "I" vowel of the compound "shrI" which again is lost in the middle of the Adi tALam. And so on. deshAdi apparently comes to us from the bhajanai tradition. tyAgarAja has used it every effectively in his kritis whereas it seems virtually unknown to the other composers of that era or before. Except...
I learned the shyAmA shAstri kriti ninnuvinaga (pUrvikalyANi) in deshAdi with vilOma mishra nadai (common practice is to sing it in vilOma mishra chApu). I remember going nuts keeping track of two oddities simultaneously, putting deshAdi and viloma mishra chApu. Are there any (other) instances of usage of dEshAdi before or during tyAgaraja's time?
deshAdi may have the same relationship to Adi as mishrachApu has to (tisra) triputa in that the chApu's are said to be half the length of the original tALam. Thus mishrachApu is considered to be a 3.5 beat tAlam. deshAdi is similarly a 4 beat tALam. Also I have read (cannot remember the source) that chApu (and deshAdi) are meant to be put briskly and not slowly. However that convention has also become blurred in practice -- even slow tempo kritis are sung with mishra chApu rather than tisra triputa. Indeed it has not been clear to me what the intended difference is between kritis of MD that are indicated in formal sources as being in mishra chApu compared to those that are in tisra tripuTa -- these days everything in seven beats is done in mishra chApu anyway.
At that time I forgot to ask my teacher how madhyAdi would be shown. Does anyone know?