take one row, compute it yourself, and you'll immediately know which formula is used.

you can also consult the original publication for phyloseq

I tested and both formulas seem to be equivalent to me or at least they yield the same result for the dummy OTU table used.

You can see below my implementation in R of the formula in the Wiki using the OTU dummy table from the other site and the result is a Bray-Curtis dissimilarity matrix equal to the other site and to phyloseq. So, I think although the formulas are different they are equivalent. This is my opinion based on the test. The implementation on the site does not check for the sum of the minimum value of each taxa/OTU across samples, neither subtract the division to one, whereas the wiki does.

bray-curtis implementation

# matrix 
otu_tbl <- matrix(
c(1, 3, 0, 1, 0, 0, 2, 0, 
  4, 4, 0, 0, 6, 2, 1),
ncol = 3,
byrow = FALSE
)
colnames(otu_tbl) <- LETTERS[1:3]
rownames(otu_tbl) <- paste0("OTU", 1:5)

bray_curtis <- function(mtx) { # assume that samples are in columns
# construct a distance mtx
samples <- colnames(mtx)
dist_mtx <- matrix(NA, nrow = length(samples), ncol = length(samples), 
                   dimnames = list(samples, samples))
# loop over a mtx i,j: 
for ( i in seq(nrow(dist_mtx)) ) { # loop over rows 
  row_samp <- rownames(dist_mtx)[i] # row sample 
  for ( j in seq(ncol(dist_mtx)) ) { # loop over cols
    col_samp <- colnames(dist_mtx)[j] # col sample
    sub_mtx <- mtx[,c(row_samp, col_samp)] # get sub mtx for samples compared
    # sum the min number across samples for each taxa
    Cij <- apply(sub_mtx, 1, function(x) min(x))
    Cij <- sum(Cij)
    # sum of the counts of sample i
    Si <-  sum(mtx[,row_samp])
    # sum of the counts of sample j
    Sj <- sum(mtx[,col_samp])
    # bray-curtis: https://en.wikipedia.org/wiki/Bray%E2%80%93Curtis_dissimilarity
    dist_mtx[i,j] <- 1 - ( 2 * Cij / ( Si + Sj ) )
  }
}
return(dist_mtx)
}

# test the function
bray_curtis(mtx = otu_tbl)
#        A         B         C
# A 0.0000000 0.6000000 0.8571429
# B 0.6000000 0.0000000 0.6842105
# C 0.8571429 0.6842105 0.0000000

# test in phyloseq
library("phyloseq")
distance(physeq = otu_table(otu_tbl, taxa_are_rows = TRUE), method = "bray")
#       A         B
# B 0.6000000          
# C 0.8571429 0.6842105

Anyway, using the phyloseq implementation of Bray-Curtis is a safe option (that relies on vegan).

I hope this helps,

António