What's the difference between Barycenter Centrality and Closeness Centrality (Freeman)? What's the difference between Barycenter Centrality and Closeness Centrality (Freeman)?

Barycenter scores are calculated as 1 / (total distance from vertex v to all other vertices) in a strongly connected network. source

Closeness Centrality (Freeman) are calculated  total distance from a node v to all the other nodes in a network. source

It looks like the same centrality .

Ps: 
I want to analyze closeness for my protein–protein interaction network.
First, I'll observe that Freeman centrality as you define it is the reciprocal of the Barycenter centrality as you define. It looks like people often use the reciprocal instead of the original definition, so in that sense they do look the same.
I found the following comparison here (though it doesn't name it as Freeman):

There are 2 types of distance centrality scores, Closeness Centrality
and Barycenter Centrality. Barycenter Centrality for vertex v defined
as:
1 / (total distance from v to all other vertices)
Closeness scores are calculated using the formula 1 / (average
distance from vertex v to all other vertices) and Barycenter scores
are calculated as 1 / (total distance from vertex v to all other
vertices).

So it appears that for this package it is a difference of the reciprocal of the total vs. reciprocal of the average.
Notably, wikipedia differs here, in that it matches your source:

Freeman's closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example.citation

Yet another source notes the existence of a &quot;normalized&quot; version of Freeman's closeness centrality, which normalizes centrality of a node by the degree of the network. It also comments on whether it is the reciprocal or not:

Closeness (Freeman '79)
The graph-theoretic distance of a given node to all other nodes. The sum of the rows/columns of the geodesic distance matrix of a graph.
Simple closeness is an inverse measure of centrality: the larger the numbers, the more distant an actor is, and the less central. Should really be called &quot;farness&quot;.
Normalized version divides the minimum &quot;farness&quot; possible (N-1) by &quot;farness&quot; to simultaneously make the range 0 to 1 and invert the measure so that larger values correspond to greater centrality (truly &quot;closeness&quot;)

So I guess that it might depend on what tools you are using. Most of the time, it might be the same. Other times, it might be normalized in different ways with either averages or for 0-1 ranges.
Closeness centrality is probably the more standard term, so it is unlikely that anyone would criticize it. But it appears that there is sometimes some variation in what people mean by that term and whether they are using a normalized or a raw closeness centrality.
In my opinion the normalization leading to a range from 0-1 is nice because it makes the measure at least somewhat generalizable across networks. But it's up to what you need to do.