Assalaamu alaikum, as you know, making Dua by directly calling on Allah is the best way. This hadith shows that you can ask Someone who is present to make Dua for you as Umar did. This is the same as one of us saying "brother please make Dua for me".
However, this hadith also beautifully shows that you cannot ask the dead or call on someone who is not able to make Dua for you. Clearly the sahaba radhiallaahu anhum believed that the prophet sallallaahu alaihi wa sallam is superior to Al'Abbas yet they chose NOT to call on him. They knew that he cannot hear their Dua for this draught or supplication for them in this. Hence they asked Abbas who was a pious servant.
Think brothers, if the sahaba believed that the prophet was present of "hazir wa nazir" or it is permissible to say "yaa Muhammad..." then why did they stop calling on him. The hadith also shows this was done on Multiple occasions, as Anas says "whenever... Used to" ie more than once. Therefore this was the practice of the sahaba to ask someone who is present and also capable of making Dua for you just as one of us today may ask someone present to make Dua for him/her.
Tawassul means nearness and yes this is a form of permissible Tawassul done by the sahaba to gain nearness to Allah by asking Abbas to make Dua. Notice they did not simple say "O abbas" while he was absent neither did they say "O Allah give us rain by the beloved position of Abbas" while he was absent.
UMAR clearly asked Abbas to make Dua for them and this by its very nature shows the impermissibility to ask anyone who is dead eg "yaa abdulqadir Jilani" ," yaa ali" etc. What the sahaba have demonstrated is enough for us.