"[T]he specific embodiment is easier to defend and has far higher chances of being approved." These are the main reasons why registered patent attorneys and agents include in just about every patent application a number of narrower (dependent) claims, which incorporate by reference and layer additional elements or further limitations onto the broadest (independent) claims. In the US Patent Office, the basic utility patent filling fee entitles you to file up to three independent claims and up to twenty claims total. An increasing additional fee applies to independent claims in excess of three or total claims in excess of twenty. In addition, as noted in a previous answer, you will have the opportunity to modify the claims during the examination (also called "prosecution") of your application, by amending, canceling, and/or adding new claims to the listing. To provide support for such amendments, it is also important to submit a detailed and thorough description of a wide range of embodiments of the invention at the outset, as you will not be allowed to claim embodiments that you did not adequately describe in the original application.
This answer is not intended to be legal advice in your particular case or to form an attorney-client relationship, and I highly recommend that you consult a patent attorney or agent.