I never used this software before but it was free trial so I had a go...
1. drag a regular speech bubble onto the page
2. drag a extend speech bubble from the bottom objects palette (?) and drop it onto the first bubble on the page
3. select the extend bubble on the page
4. drag it away. the further it goes the skinnier the joining connection appears
5. increasing the size of each bubble seems to make the joiner connect point skinnier (even more so than just a relative size, it seems to get absolutely smaller...)
I dunno if that's what you wanted...
EDIT: HODOR HODOR
Okay i re-read what you wrote... and I can reproduce the issue is you described: using the "Inspector" window, you can't make the connector "thin" and the tail on the first bubble to be "medium" thick... because the setting for tail width is applied to both, not each tail part separately.
Looks like a limitation in the current version of the app - and it's weird because some other properties *are* bubble specific. But the tails seem to apply to both connected bubbles.
Maybe the creator of the comic you used as an example did some photoshop work? i.e. did not make extended bubbles, but used a "normal" bubble as the second one, hid the tail, and then added the connection as post-work?