It is the 3rd and 7th day after touching the dead corpse
That seems somewhat implied from the context of the verses you quote. I honestly would not have ever thought to consider Tuesday/Saturday, but in thinking about your question, I could see how someone might question it (though days of the week are not really mentioned in context).
However, the following verses make it clear it is from touching the corpse.
I believe you are using the NIV in your question, so here is that version both with your original verses and some of the following verses, Num 19:11-19 in total (emphasis added), with commentary between:
11 “Whoever touches a human corpse will be unclean for seven days.
12 They must purify themselves with the water on the third day and on
the seventh day; then they will be clean. But if they do not purify
themselves on the third and seventh days, they will not be clean.
13 If they fail to purify themselves after touching a human corpse,
they defile the LORD’s tabernacle. They must be cut off from Israel.
Because the water of cleansing has not been sprinkled on them, they
are unclean; their uncleanness remains on them.
So vv.11-13 declare three critical things to answering the question: (1) they will be unclean for a total of 7 days; (2) they are considered clean once purification has occurred on the 3rd/7th day; (3) this purification involves a sprinkling on them.
14 “This is the law
that applies when a person dies in a tent: Anyone who enters the tent
and anyone who is in it will be unclean for seven days, 15 and every
open container without a lid fastened on it will be unclean.
16 “Anyone out in the open who touches someone who has been killed
with a sword or someone who has died a natural death, or anyone who
touches a human bone or a grave, will be unclean for seven days.
vv.14-16 define further what essentially qualifies as "touching" a dead person (which can include just entering the tent where they died).
17 “For the unclean person, put some ashes from the burned
purification offering into a jar and pour fresh water over them.
18 Then a man who is ceremonially clean is to take some hyssop, dip it
in the water and sprinkle the tent and all the furnishings and the
people who were there. He must also sprinkle anyone who has touched a
human bone or a grave or anyone who has been killed or anyone who has
died a natural death. 19 The man who is clean is to sprinkle those who
are unclean on the third and seventh days, and on the seventh day he
is to purify them. Those who are being cleansed must wash their
clothes and bathe with water, and that evening they will be clean.
vv.17-19 give the final conclusion that definitively answers the question. This process of purifying is being done by one who is still clean (so it is not the individual themselves doing it) as a sprinkling. The clean person does this to the unclean on the 3rd/7th days, but note the last statement of v.19--the individual being cleansed must wash and bath that final day, and "that evening they will be clean." Now coupled with v.11 stating that uncleanness would last 7 days, then the 7th day cleansing has to be in relation to the time from touching the dead, because after the 7th day washing and this bathing, the person is considered clean.
If it meant Tue/Sat, then one touching the corpse on Friday would cleanse Saturday and Tuesday, and be clean, but will not have been unclean for 7 days (not matching v.11). So the count must be from the time of touching the corpse.
Of "third day" (etc.)
To answer your further question in your comment: No, the Hebrew is the same, just like the English is the same—"third day" of the week, "third day" since I touched a dead corpse, etc.
The Hebrew word "third" is שְׁלִישִׁי (shelîshî) the ordinal form of three, and "day" is יוֹם (yôm). Which third day counted from what reference is based on context, just like English. So the same words are found in Gen 1:13 of the third day of creation week, Gen 22:4 of the third day of Abraham's travels, 2 Kg 20:8 of the third day from the time of prophesying Hezekiah's healing, etc.