Are two claims considered different if they both contain the same
limitations but one of them recites some of the limitations in the
preamble?

For US Law

Yes, they are different.  The preamble DOES NOT limit the claim,
unless it is a Jebson claim where the preamble is reciting the prior
art.

The claim limitations are those words after "consisting", "comprising"
or other similar terms.

I did a little research and found a few cases which state that a
claim?s preamble can be considered a claim limitation when it is
?necessary to give life, meaning and vitality? to the claim.  See In
Re Cruciferous Sprout Litigation, 2002 WL 1917373 (Fed. Cir. 2002)
(citing Corning Glass Works v. Sumitomo Elec. U.S.A., Inc., 868 F.2d
1251, 1257 (Fed. Cir. 1989).  On the other hand, if the preamble
?offers no distinct definition of any of the claimed invention?s
limitations, but rather merely states . . . the purpose or intended
use of the invention,? then the preamble cannot be read as a
limitation on a claim.  See Pitney Bowes, Inc. v. Hewlett-Packard
Co.,182 F.3d 1298, 1305 (Fed. Cir. 1999).
Is this research flawed?  I am just a student.  Thanks.