Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally defined? References would be lovely. Best regards
Profinite groups were first called "Groups of Galois type", see J.P. Serre's book "Cohomologie Galoisienne" of $1964$. The term "profinite" comes from Serre (if I am not mistaken). Of course, some profinite groups have a much older history, e.g., already Hensel defined in $1910$ the $p$adic integers during his studies of algebraic numbers. As to the definition, a profinite group is a Hausdorff, compact, and totally disconnected topological group. The other (equivalent) definition, better adapted to the name "profinite", is, that a profinite group is a group which is isomorphic to the inverse limit of an inverse system of discrete finite groups. 


Tate was the first to study the cohomology of profinite groups. In his unpublished 1958 article (reprinted as Chapter VII of Lang's notes "Topics in the Cohomology of Groups"). He writes: "We introduce a new category of groups and a cohomological functor, obtained as limits from finite groups. A topological group G will be said to be of Galois type if it is compact, and if the normal open subgroups form a fundamental system of neighborhoods of the identity e." He later points out that "every group of Galois type is the inverse limit of its factor groups G/U taken over all open normal subgroups. Thus one often says that a Group of Galois type is profinite." Serre's "Cohomologie Galoisienne" is the notes of his 196263 course. As Serre notes (end of Chapter I), "almost all the results of Sections 1,2,3,4 are due to Tate, who published nothing". At the end of Chapter II, Serre writes "The situation is completely analogous to that of Chapter I: almost all the results are due to Tate." Serre is often mistakenly credited with these results by careless readers. 


See also W.KrullGaloische theorie der unendlichen algebraischen erweiterungenMathematische Annalen,100,1928 The concept is there,maybee not the name 

