There are several permutations to your important question: defining "ethnic" vs. "religious" Judaism, from both historical and contemporary perspectives. While the term of "religious" can be ambiguous, I will define it here as being "Jewish" in accordance with Jewish law as opposed to "ethnic" Judaism which would have implications for social integration or personal identification.  
The distinction between "religious" and "ethnic" Judaism exists in certain parts of our history, though not in others. Before the Torah was given, Jacob's son Judah married an unnamed Cannanite woman only known as the daughter of Shua (Gen. 38:2), and the while the children of this marriage 'Er and Onan died, they were still recognized as being Judah's children (Gen. 46:12). Joseph's children Menashen and Ephraim were born from an Egyptian woman (Gen. 41:50), yet merited a special blessing from Jacob and were treated as "tribes" in their own right (Gen. 48).  In these examples, the "Jewishness" was determined by the father, but predating the giving of the Torah, it may make more sense to view this Jewishness as more "ethnic" than "religious" as it pertains more to familial attachments.
Since at least the times of the Mishnah (M. Kiddushin 3:12), religious Jewishness was defined as someone being born of a Jewish mother, or having converted in accordance with Jewish Law.  This definition has endured for millennia such modern reformations to define Jewishness based on the father's status were met with harsh criticism. During this extensive period, I would suggest the division between ethnic and religious Judaism was not nearly as pronounced nor as relevant as it is today, mostly because Jewish law only recognizes halakhic Jewishness, not ethnic Jewishness. (Note that this distinction is only meaningful when there is a conflict between the two).  Depending on the location and century, outside forces such as governmental segregationist policies contributed towards the conflation of ethnicity and religion. 
We are probably more familiar with the ethnic/religious dichotomy as it pertains to contemporary identity, and this is where there is much personal confusion. Once Jews gained the freedom to assimilate, being "Jewish" became more of a conscious decision. Therefore it is not as unusual to find someone claiming to be Jewish based on personal identity rather than meeting halakhic criteria. However, personal Judaism may not even be classifiable as "ethnic" Judaism if membership is conditional on communal acceptance. If we view Judaism as a purely social phenomenon, then acceptance into the tribe would be predicated on other members of the "tribe" validating an individual's identity.  This is a much harder question to answer historically given that we are limited by the data available. Even if we can produce evidence of one group accepting another's Jewishness we cannot be assured other group would not have rejected an individual. Not being a historian, I cannot responsibly offer any speculations in this regard for any time period.
Today we can easily identify which groups accept and reject which definitions of Jewishness, both in terms of ethnicity, identity, or Jewish law. For obvious reasons, this fundamental question of identity and belonging remain highly contentious.
 

The idiom in Hebrew to which this question is referring is avoda zara, which while normally translated as "idolatry" literally means "strange" or "foreign" worship. Thus while the idiom of avoda zara most often refers to worshiping other Gods or spirits, the "strangeness" may also be applied to unauthorized worship of the Jewish God. Focusing on ritual practices outside of Judaism, Jewish law forbids participating in avoda zara rituals regardless of one's personal intent as well as performing acts of worship which are not typical of that religious tradition (M. Sanhedrin 7:6, B. Sanhedrin 60b).

 

There are two components to this question which are as essential as they are controversial. The first and most obvious point to consider is how is avoda zara defined today in practical terms as it relates to other religious practices. Some cases will be more apparent. For example, taking communion in a Catholic Church, where the ritual is defined as eating the body of their savior, would seem to me to be under the category of performing an avoda zara ritual, even if one's intention is only to fulfill some personal satisfaction. Where the question becomes more ambiguous is precisely regarding the sorts of rituals described here. In the Talmud, R. Yochanan requires members of the Jewish High Court to be "experts in sorcery" in order to properly adjudicate these sorts of decisions (B. Sanhedrin 17a), indicating the complexity of such questions. Unfortunately, from what I have found in my own research, I am unable to offer a clear unified theory as to what is included or excluded, or a formula for evaluating new practices.

Complicating matters further is the halakhic connection between "practices and ideas." While practices may be forbidden, thoughts cannot be actionable. At best, one can be "megaleh da'at" - reveal one's intentions - but this requires an act of speech in which one declares one's thoughts. Regarding the thoughts themselves, one Talmudic passage states that the thoughts of sinning can be harder than the sin itself (B. Yoma 29a), while another posits that God does not equate thoughts with actions regarding punishments (B. Kiddushin 39b). Thus incorporating a foreign "idea," especially when kept privately, may not carry the same halakhic implications as performing practices.

But while I cannot provide a definitive answer to the question, I would like to share my own personal guide, which I have adapted from my teachers. Judaism has its own share of what can be called "superstitions" ranging from "segulah wine" at weddings to kapparot before Yom Kippur, itself a practice which R. Yosef Karo considered pagan (Beit Yosef O.C. 605). In addressing these sorts of rituals, one of my teachers suggested that the more seriously one perceives these extra-religious practices to be effective, the more problematic they become. Perhaps this rationale could be applied to the more innocuous superstitions one encounters. For example, while I do not think it is prohibited to skim the daily horoscope, I would personally find it more problematic if people were to plan their lives around it. Note that this is only an approach, not an answer, and that I would suggest that each case needs to be evaluated separately.

Finally, I would be remiss if I did not suggest a degree of introspection as to why someone finds these beliefs or ideas attractive. I understand that spiritual journeys often detour in unexpected directions, but I also know that the Jewish religious tradition is exceptionally vast and eclectic. For whatever questions one is seeking answers or insight, I would not preclude also seeking from texts or teachers from within the Jewish tradition.