Is the finite automaton for 0*1*0*0 the same as the one for 0*1*0+, or for 0*1*00*?

Have come across a question 1.e here that draws the finite automaton for: 0*1*0*0, but instead draws one for: 0*1*00* or 0*1*0+.

Am confused by the same, as that means: 0*0 = 00*; which though the same semantically; but still confuses somewhat.

Any theorem to explain the same, or it is just a lack of practice?