1.) The collector-base voltage for T1 is Vcb=0V. Hence, you are just at the threshold for the active mode.
2.) In case of T2 the base-collector diode also is open allowing an additional current through the base node. The transistor cannot work as an amplifier and Ib(T2)>Ib(T1).
EDIT 1 : For clarification of some misunderstandings: The OP did ask for the base current Ib - that is the current through the base node, which NOT necessarily is identical with the current through the base-emitter path. In case the base-collector junction is forward biased (as in cas 2 with Vce=0.1 V) the base current Ib has two components: Ibe and Ibc.
EDIT 2 : This effect (current from base to collector) can be clearly verified for Vce=0. For this case, all detailed Ic=f(Vce) graphs show that for Vce=0 all curves start at NEGATIVE Ic values (they do not start at the origin). And negative Ic values describe a current out of the collector node (npn case). This is nothing else than Ibc as part of the current Ib=Ibe+Ibc.