Looking at the documentation you provide, I think that I can provide definitions for those terms as defined by EXPASY. Note that those definitions may just be an in-house system that EXPASY uses that you can't generalize to other databases for instance.
First, in section I.C of the documentation, they write "The use of protein sequence patterns (or motifs) to determine the function(s) of proteins is becoming very rapidly one of the essential tools of sequence analysis."
So it looks like for them patterns = motifs and motifs = patterns. Note again that "pattern" appears to be a somewhat informal term. I would certainly say that it is confusing that they use them as synonyms, but they do at least define them as such.
Second, I think that they are using the word "profile" in a very narrow technical sense, with regard to the way that they detect homology between proteins. The standard way to detect homology between proteins is using programs called Hidden Markov Models (HMMs). In section II.B, they write "A profile or weight matrix (the two terms are used synonymously here)...". So a profile is really just a weight matrix derived from a HMM that can be applied to any given alignment to figure out whether or not it matches homology. They go on to give an example of how this weight matrix is applied.
In simple terms, how the HMM works is that you give it a bunch of sequences that you know belong to the same family, and it will try to learn the frequencies of amino acids at each position and what order they come in. This model of the amino acid frequencies is the weight matrix they talk about. Then you make sure that the HMM works well (gives high scores for proteins in the same family, low scores for proteins in different families), and then you can apply the program to try to find new homologs.
Hope that answers the Q.