Those are all different ways of configuring hopw the shipping rate will be applied.
the triggers section is used to determine the conditions that must be met for the shipping rat to apply, for example, in oyur case, you would want top trigger the shipping rules based on the "eCartCheckoutForm_shipping_country" session value.
for the UK rule, you would set the trigger as:
based on sesson variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "=" to "UK"
for the rest of Europe, but not UK, you would use the same trigger, using the country codes for the other European countries:
based on session variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "=" to "FR"
AND
based on session variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "=" to "SP"
etc
for the rest of the world, you would use Session variable "eCartCheckoutForm_shipping_country" is set
AND
based on session variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "!=" to "UK"
AND
based on session variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "!=" to "SP"
AND
based on session variable value:
if the session variable "eCartCheckoutForm_shipping_country" has a value "!=" to "FR"
etc...
if the rules are not sticking, most likely the eCart object file is set to be read only, make sure all the files in the WA_eCart folder are not set to be read only.
Shouldn't it be OR instead of AND?