The Russian Orthodox way (which so far as I experienced, involves the former) is cool, but as a non-member I an relegated to taking the Eucharist the mainstream Catholic way.
The former is an okay description of how we do it. What we do specifically, for the information of those in this thread, is cut the bread into small pieces and suspend them in hot wine, with the priest spoonfeeding it to the faithful. I like the Orthodox way of doing it and prefer BRTD's way from the options given, because it reminds us of Paul's exhortation to the Corinthians, "For we, though many, are one bread and one body; for we all partake of that one bread."
My first time at an ROC service I mistakenly took communion because, per my recollection, Catholics were supposed to be able to take the Eucharist anywhere.
At your church, is there a small table one approaches after receiving communion, at which there is additional bread and wine? Such was my experience and it struck me as rather odd.