However, the indefinite conjugation is used when the direct object is a first or second person pronoun.
As a native speaker (not a linguist) I disagree that we should call this "indefinite conjugation". Let us take the verb szeret = love as an example, and conjugate it for a first person subject.
- szeretek (valamit) = I love (something) (indefinite 3rd person object)
- szeretem (őt) = I love him/her/it (definite 3rd person object)
- szeretlek (téged) = I love you (2nd person object)
Notice that the verb ending used for the 2nd person object is different from either the definite or indefinite 3rd person objects. Thus I see no basis for calling it indefinite. Parts that can be omitted are in parentheses: there is usually no need to state the object in Hungarian, as it's clear from the verb ending.
For completeness, let me also list the conjugation for a 2nd person subject:
- szeretsz (valamit) = you love (something) (indefinite 3rd person object)
- szereted (őt) = you love him/her/it (definite 3rd person object)
- szeretsz (engem) = you love me (1st person object)
Thus it is true that in this case the 1st person object uses an ending of the same form as the 3rd person indefinite object. Does that mean that we should consider them to be the same ending? Of course not—we can call them homophones. Were the endings the same in the distant past as well? I do not know.
A quick search for this topic yielded the following article:
A very brief summary is that originally only definite objects were marked with a suffix (-t, same as today) on the object itself. Indefinite objects were not marked. Thus it was clear from the form of the object alone whether it was definite or indefinite. At some point, indefinite objects started to get marked with -t as well, and verb conjugation took over the role of indicating object definiteness. At first, an analytic structure (verb + subject pronoun) was used to indicate definiteness. This later evolved into the suffixes we use for 3rd person definite objects. The 2nd person object form is said to have a different origin, but the article does not discuss it in detail.
As you can see from my original example, we do not have a complete set of endings to denote every combination of subject and object person/number. The bottom line of the article is that the reason for this is not that the two conjugations are gradually collapsing (from an older completely separate form), but that they have never fully separated during the evolution of the language.