Leviticus 17:3-4 is talking to the Israelites in the desert when they were all camping together and in close proximity to the tabernacle, or even later in the period of the Judges where the Israelites were still tribal and in close proximity to each other. They were then forbidden to eat meat unless they sacrificed the inedible parts to the Lord in the tabernacle. The impression is indeed, as the OP surmised, that any slaughter outside the temple is forbidden.
Deuteronomy 12:15-16 is focused on a later period, long after the Israelites settled in Canaan and spread out south and north of Jerusalem, at that point not everyone lived in close proximity to the temple, so Leviticus' ancient law wasn't anymore practical, so the law had to be revised somewhat, this is evident from v. 20-23:
When the Lord your God has enlarged your territory as he promised you,
and you crave meat and say, “I would like some meat,” then you may eat
as much of it as you want. 21If the place where the Lord your God
chooses to put his Name is too far away from you, you may slaughter
animals from the herds and flocks the Lord has given you, as I have
commanded you, and in your own towns you may eat as much of them as
you want. 22Eat them as you would gazelle or deer. Both the
ceremonially unclean and the clean may eat
This is also the meaning of v. 15
However, you may slaughter and eat meat within any of your gates,
whatever you desire, according to the blessing of the Lord your God
which He has given you; the unclean and the clean alike may eat it, as
the gazelle and the deer.
It's referring to the same period when the Israelites were spread out, the bible is saying here that the law in Leviticus is not applicable anymore once their territory has been enlarged. However, it still applies in the case when one is bringing it for sacrificial purposes; if it is intended as a sacrifice then the law of Leviticus is still in place and he must bring up to the Temple mount for sacrifice. This is enumerated in the verses before v. 15:
Be careful not to sacrifice your burnt offerings anywhere you please.
14Offer them only at the place the Lord will choose in one of your
tribes, and there observe everything I command you. (v. 13-14)
This I believe is the proper reconciliation between Deut. and Lev. I believe that @Tonychan's answer is only partially correct here, as I have noted in my comments.
Hope this helps.