does the classification of your Bachelor degree really matter that much?

If you are talking about postgraduate study or getting your first job then the answer is yes. PhD programmes require you to have a 2(i), or a 2(ii) plus a masters degree. Many Masters degree programmes restrict entry according to the grade of your first degree.

When it comes to getting your first job your degree grade may be all the employer has to go on (very few people are employed because they ‘like reading, cinema, going out and tennis, and once hitch hiked round Thailand with a goat’). After you have work experience or a postgraduate qualification then your first degree grade will become less significant, but in times when nearly 50% of school leavers go to university simply having a degree doesn't make you much different from everyone else - you need to get the best one you can.