Self-made Schottky transistor doesn't behave as I'd expect

Your Schottky diode is in lieu of a Baker clamp, which is like this:

^{simulate this circuit – Schematic created using CircuitLab}

and the diodes are switch diodes; that 1N5819 diode is a rectifier, which has higher current (and stray capactance) than a switch. The 1N5819 is about 110 pF of capacitance, while a 1N4148 switch is 4 pF.

Circuit #2 might be showing the benefit of paralleling the base resistor with a 100 pF capacitor, which speeds up transistor turnon and turnoff, rather than the clamp benefit of reducing base charge storage. Your circuit #3 might be showing the capacitance C-B causing that ramp, so a change of component, from Schottky rectifier 1N5819 to Schottky switch 1N5711, about 2 pF, should help. A small capacitor across R1 is a good bet, too.