Is a busker/streetsinger a folk musician?

I think so. You may differ.

When deciding upon a repertoire, a person trying to make some money singing and self-accompanying on the public street will want to include selections that are:

Recognizable to the public;
Enjoyable to sing and to hear;
Meaningful to oneself, to make a long day and/or night as endurable as possible;
Able to be recognizably performed using the portable (and usually acoustic) instrument of one's choice.

Many songs that may not fit the strictest definition of "traditional/folk" make excellent selections for street performers. Many Beatles songs, obviously enough, would qualify. Quite a few other Beatles songs would be very likely NOT make the cut, notably those late-career compositions whose very identity is tied up with electronic studio-produced sounds (e..g., "I Am the Walrus," etc.)

I assert, quite firmly, that "buskable" songs are, ipso facto, folk songs for today, the embodiment of a living tradition.

Of course, there are many who believe that only those songs that represent a long-gone historical tradition are "true" folk songs. Everyone is certainly entitled to an opinion. Whether a repertoire composed exclusively of such songs will actually appeal to living listeners is an interesting question.

The skill and talent of the performer, in the end, will usually determine the music's success. If a singer truly and passionately believe that the material he/she chooses to present (old or new or both) is uniqely worthwhile, the performance will be valid and appealing and certainly should find its own audience.