That is not the easiest question in the world, you are both right and wrong depending on the time frame in which you look at the Tiger naming conventions.

According the following books this is the naming convention at different moments in time:

Osprey - New Vanguard 5 - Tiger I heavy tank ISBN 1-85532-337-0

page 16 - Official designations

as of 2 March 1942: Panzerkampfwagen VI H (8.8 cm) (SdKfz 182) Ausfuhrung H1.

as of 5 March 1943: Panzerkampfwagen Tiger (8.8 cm L/56) (SdKfz 181) Ausfuhrung E.

Osprey - New Vanguard 1 - Kingtiger heavy tank ISBN 1-85532-282-X

page 11 - Official designations

as of 16 March 1943 Tiger II instead of Tiger H3

as of 2 Jun 1943: Panzerkampfwagen Tiger (8.8 cm) (SdKfz 182) Ausf B

Note that the Tiger I was 1st named Sonderkraftfahrzeug 182 and later renamed to Sonderkraftfahrzeug 181. And when the Tiger II came along they decided to reuse the SdKfz 182 number. So with a bit of reasoning you might say that the "ausfuhring E" was a variant of the SdKfz 181 - Tiger model and the "Ausfuhrung B" was a variant of the SdKfz 182 - Tiger II.

It probably does not make things clearer but the names have to be seen in the year/month in which the original documents were written.